PHYSICS

RECTILINEAR MOTION

BASIC CONCEPTS

Average Velocity (in an interval) : vav

= v = <v> = takentimeTotal

ntdisplacemeTotal =

t

rr if

Average Speed (in an interval) Average Speed = taken timeTotal

travelled distanceTotal

Instantaneous Velocity (at an instant) : instv

=

t

rlim

0t

dx

dy for some common functions :

y dx

dy Y

dx

dy

xn nxn–1 sec x sec x tan x sin x cos x cosec x – cosec x cot x cos x – sin x log x 1/x tan x sec2 x ex ex cot x – cosec2 x

(a)dx

d (cy) = c

dx

dy (c is a constant) (b)

dx

d (u + v) =

dx

du +

dx

dv

(c)dx

d ( uv) =

dx

duv + u

dx

dv(d)

dx

d

v

u = 2v

dxdv

udxdu

v

(e)dx

dy =

du

dy . dx

du

(f) If y = uv

then, take log on both sides.

log y = v.log(u)

Now differentiate both sides.

Average acceleration (in an interval): ava

= t

v

= t

vv if

Instantaneous Acceleration (at an instant): a

= dt

vd

=

t

vlim

0t

Graphs in Uniformly Accelerated Motion along a straight line (a 0)

x is a quadratic polynomial in terms of t. Hence x t graph is a parabola.

xi

x

a > 0

t0

xi

x

a < 0

t0

x-t graph

v is a linear polynomial in terms of t. Hence vt graph is a straight line of slope a.

v

ua is positive

slope

=a

t0

v

u

a is negative

slope = a

t0

v-t graph

at graph is a horizontal line because a is constant.

a

apositiveacceleration

t0a

a

negativeacceleration

0

a-t graph

For uniformly accelerated motion (a 0), xt graph is a parabola (opening upwards if a > 0 and openingdownwards if a < 0). The slope of tangent at any point of the parabola gives the velocity at that instant.

For uniformly accelerated motion (a 0), vt graph is a straight line whose slope gives the acceleration of theparticle.

In general, the slope of tangent in xt graph is velocity and the slope of tangent in vt graph is the acceleration.

The area under at graph gives the change in velocity.

The area between the vt graph gives the distance travelled by the particle, if we take all areas as positive.

Area under vt graph gives displacement, if areas below the taxis are taken negative.

Maxima & Minima

dx

dy = 0 &

dx

d

dx

dy < 0 at maximum and

dx

dy = 0 &

dx

d

dx

dy > 0 at minima.

Equations of Motion (for constant acceleration)

(a) v = u + at (b) s = ut + 2

1 at2 s = vt

2

1 at2 x

f = x

i + ut +

2

1 at2

(c) v2 = u2 + 2as (d) s = 2

)vu( t (e) s

n = u +

2

a (2n 1)

For freely falling bodies : (u = 0)

(taking upward as positive)

(a) v = – gt (b) s = –2

1 gt2 s = vt

2

1 gt2 h

f = h

i –

2

1 gt2

(c) v2 = – 2gs (d) sn = –

2

g (2n 1)

PHYSICS

E X E R C I S E 1DISTANCE AND DISPLACEMENT

1. A car starts from P and follows the path as shown in figure. Finally car stops at R. Find the distance travelled

and displacement of the car if a = 7 m, b = 8 m and r = 11

m? [Take 7

22 ]

2. A man has to go 50 m due north, 40 m due east and 20 m due south to reach a field.

(a) What distance he has to walk to reach the field ?

(b) What is his displacement from his house to the field?

3. A sail boat sails 2km due East, 5km 37o South of East and finally an unknown displacement . If the finaldisplacement of the boat from the starting point is 6km due East, the third displacement is ______

4. A particle P is moving with a constant speed of 6m/s in a direction k2ji2 . When t = 0, P is at a point

whose position vector is k7j4i3 . Find the position vector of the particle P after 4 seconds.

5. A hall has the dimensions 10 m × 10 m × 10 m. A fly starting at one corner ends up at a diagonally oppositecorner. The magnitude of its displacement is nearly

(A) 5 3 m (B) 10 3 m (C) 20 3 m (D) 30 3 m

6. A particle moves along a path ABCD as shown in the figure. Then the magnitude of net displacement of theparticle from position A to D is :

(A) 10 m (B) 25 m (C) 9 m (D) 27 m

7. A particle moves in a plane from A to E along the shown path. It is given that AB = BC = CD = DE = 10metre. Then the magnitude of net displacement of particle is :

E

D

C

BA

108°

108°

108°

(A) 10 m (B) 15 m (C) 5 m (D) 20 m

8. When a person leaves his home for sightseeing by his car, the meter reads 12352 km. When he returnshome after two hours the reading is 12416 km.

(a) What is the average speed of the car during this period ?

(b) What is the average velocity?

9. A clock has a minute-hand 10 cm long. Find the average velocity between 6.00 AM to 6.30 AM for the tip ofminute-hand.

(A) 21

22 cm min–1 (B)

21

2 cm min–1 (C)

21

12 cm min–1 (D)

3

2 cm min–1

10. A particle covers each 3

1 of the total distance with speed v

1, v

2 and v

3 respectively. Find the average speed of

the particle ?

11. A boy start towards east with uniform speed 5m/s. After t = 2 second he turns right and travels 40 mwithsame speed. Again he turns right and travels for 8second with same speed. Find out thedisplacement; average speed, average velocity and total distance travelled.

12. A car travels from A to B at a speed of 20 km h–1 and returns at a speed of 30 km h–1. The average speed ofthe car for the whole journey is :

(A) 5 km h–1 (B) 24 km h–1 (C) 25 km h–1 (D) 50 km h–1

13. A person travelling on a straight line moves with a uniform velocity v1 for some time and with uniform velocity

v2 for the next equal time. The average velocity v is given by

(A) 2

vvv 21 (B) v 21 vv (C)

21 v

1

v

1

v

2 (D)

21 v

1

v

1

v

1

14. A body covers first 3

1 part of its journey with a velocity of 2 m/s, next

3

1 part with a velocity of 3 m/s

and rest of the journey with a velocity 6m/s. The average velocity of the body will be

(A) 3 m/s (B) 3

11m/s (C)

3

8 m/s (D)

3

4 m/s

15. One car moving on a straight road covers one third of the distance with 20 km/h and the rest with 60km/h. The average speed of the car is

(A) 40 km/h (B) 80 km/h (C) h/km3

246 (D) 36 km/h

16. A car runs at constant speed on a circular track of radius 100 m taking 62.8 s on each lap. What is theaverage speed and average velocity on each complete lap?

(A) velocity 10m/s, speed 10 m/s (B) velocity zero, speed 10 m/s

(C) velocity zero, speed zero (D) velocity 10 m/s, speed zero

17. An ant is at a corner of a cubical room of side ' a

'. The ant can move with a constant speed u. The

minimum time taken to reach the farthest corner of the cube is :

(A) u

a3(B)

u

a3(C)

u

a5(D)

u

a12

18. The position of a body is given by x = At + 4Bt3, where A and B are constants. Find (a) the dimensions of Aand B, (b) acceleration as a function of time, (c) velocity and acceleration at t = 5 s.

19. Find the velocity as a function of time if x = At + Bt–3 , where A and B are constants. ?

20. An athelete takes 2s to reach his maximum speed of 18 km/h. What is the magnitude of his averageaccleration?

21. The displacement of a body is given by 2s = gt2 where g is a constant. The velocity of the body at any timet is:

(A) gt (B) gt/2 (C) gt2/2 (D) gt3/6

EQUATIONS OF MOTION AND MOTION UNDER GRAVITY

22. A car accelerates from 36 km/h to 90 km/h in 5 s. what was its acceleration in m/s2. and how far did it travelin this time? Assume constant acceleration.

23. A train starts from rest and moves with a constant acceleration of 2.0 m/s2 for half a minute. The brakes arethen applied and the train comes to rest in one minute. Find (a) the total distance moved by the train, (b) themaximum speed attained by the train and (c) the position(s) of the train at half the maximum speed.

24. A particle moving along a straight line with constant acceleration is having initial and final velocity as 5 m/sand 15 m/s respectively in a time interval of 5 s. Find the distance travelled by the particle and the accelerationof the particle. If the particle continues with same acceleration, find the distance covered by the particle inthe 8th second of its motion.

25. A car travelling 72 km/h deccelerates uniformly at 2 m/s2. Calculate (a) the distance it goes before it stops,(b) the time it takes to stop, and (c) the distance it travels during the first and third seconds.

26. A ball is dropped from a tower. In the last second of its motion it travels a distance of 15 m. Find the heightof the tower. [take g = 10m/sec2]

27. A particle is moving with initial velocity kjiu

. What should be its acceleration so that it canremain moving in the same straight line ?

(A) k2j2i2a

(B) k2j2i2a

(C) k3j3i3a

(D) j1i1a

28. A particle has a velocity u towards east at t = 0. Its acceleration is towards west and is constant, Let xA and

xB be the magnitude of displacements in the first 10 seconds and the next 10 seconds.

(A) xA < x

B(B) x

A = x

B(C) x

A > x

B

(D) the information is insufficient to decide the relation of xA with x

B.

29. A body starts from rest and is uniformly acclerated for 30 s. The distance travelled in the first 10 s is x1, next

10 s is x2 and the last 10 s is x

3. Then x

1 : x

2 : x

3 is the same as

(A) 1 : 2 : 4 (B) 1 : 2 : 5 (C) 1 : 3 : 5 (D) 1 : 3 : 9

30. A ball is dropped from the top of a building. The ball takes 0.5 s to fall past the 3 m length of a window somedistance from the top of the building. If the speed of the ball at the top and at the bottom of the window are v

T

and vB respectively, then (g = 9.8 m/sec2)

(A) vT + v

B = 12 ms–1 (B) v

T – v

B = 4.9 m s–1 (C) v

Bv

T = 1 ms–1 (D)

T

B

v

v = 1 ms–1

31. A stone is released from an elevator going up with an acceleration a. The acceleration of the stone after therelease is

(A) a upward (B) (g-a) upward (C) (g-a) downward (D) g downward

32. The initial velocity of a particle is u (at t=0) and the acceleration f is given by (f = at). Which of thefollowing relations is valid?

(A) v = u + at2 (B) v = u + 2

at2

(C) v = u + at (D) v = u

PHYSICS33. A stone is dropped into a well in which the level of water is h below the top of the well. If v is velocity of

sound, the time T after which the splash is heard is given by

(A) T = 2h/v (B) v

h

g

h2T (C) v2

h

g

h2T (D) v

h2

g2

hT

34. A student determined to test the law of gravity for himself walks off a sky scraper 320 m high with astopwatch in hand and starts his free fall (zero initial velocity). 5 second later, superman arrives at thescene and dives off the roof to save the student. What must be superman's initial velocity in order thathe catches the student just before reaching the ground ?

[Assume that the superman's acceleration is that of any freely falling body.] (g = 10 m/s2)

(A) 67.23 m / s (B) 91.66 m / s (C) 102.91 m/s (D) It is not possible

35. In the above question, what must be the maximum height of the skyscraper so that even supermancannot save him.

(A) 65 m (B) 85 m (C) 125 m (D) 145 m

36. A particle is moving along a straight line with constant acceleration. At the end of tenth second its velocitybecomes 20 m/s and in tenth second it travels a distance of 10 m. Then the acceleration of the particle willbe-

(A) 10 m/s2 (B) 20 m/s2 (C) 5

1m/s2 (D) 3.8 m/s2

37. In the figure shown ABCD is a rectangular smooth tube kept fixed in a verticalplane. A particle is projected from point A to reach point C with some speed.At the corners B and D velocity changes its direction by 90º without anychange of its magnitude at that corner. If time taken on paths ABC and ADCare t

1 and t

2 respectively, then: (given > b)

(A) t1 = t

2(B) t

1 > t

2(C) t

1 < t

2(D) none of these

GRAPHS RELATED QUESTION

38. For a particle moving along x-axis, velocity-time graph is as shown in figure. Find the distance travelled anddisplacement of the particle? Also find the average velocity of the particle?

39. The acceleration of a cart started at t = 0, varies with time as shown in figure . Find the distance travelled in30 seconds and draw the position-time graph.

40. Two particles A and B start from rest and moves for equal time on a straight line. The particle A has anacclereation a for the first half of the total time and 2a for the second half. The particle B has an acceleration2a for the first half and a for the second half. Which particle has covered larger distance?

PHYSICS41. Figure shows position-time graph of two cars A and B.

A

0

5

x(m)

B

(A) Car A is faster than car B. (B) Car B is faster than car A.

(C) Both cars are moving with same velocity. (D) Both cars have positive acceleration.

42. Fig. shows the displacement time graph of a particle moving on the X-axis.

(A) the particle is continuously going in positive x direction

(B) the particle is at rest

tot

x

(C) the velocity increases up to a time to, and then becomes constant.

(D) the particle moves at a constant velocity up to a time to, and then stops.

43. The displacement–time graph of a moving particle is shown below. The instantaneous velocity of the particleis negative at the point :

t

x

C

D

E F

(A) C (B) D (C) E (D) F

44. The velocity time graph of a particle moving along a straight line in a given time interval is as shown infigure. Then the particle (with increase in time starting from t = 0 sec.)

(A) speeds up (B) speeds down

(C) first speeds down and then speeds up (D) first speeds up and then speeds down

45. The variation of velocity of a particle moving along a straight line is shown in the figure. The distance travelledby the particle in 4 s is :

(A) 25 m (B) 30 m (C) 55 m (D) 60 m

PHYSICS46. A particle starts from rest and moves along a straight line with constant acceleration. The variation of velocity

v with displacement S is :

(A)

S

v

(B)

v

S

C)

v

S

(D)

v

S

47. The displacement time graphs of two particles A and B are straight lines making angles of respectively 300

and 600 with the time axis. If the velocity of A is vA and that of B is v

B, then the value of

B

A

v

v is

(A) 2

1(B)

3

1(C) 3 (D)

3

1

48. The drawing shows velocity (v) versus time (t) graphs for two cyclists moving along the same straightsegment of a highway from the same point. The first cyclist starts at t = 0 min and the second cycliststarts moving at t = 3.0 min. The time at which the two cyclists meet is : (Both velocity-time curvesintersect at t = 4 min)

(A) 4.0 min (B) 6.0 min (C) 8 min. (D) 12 min.

49. Car A and car B move on a straight road and their velocity versus time graphs are as shown in figure.Comparing the motion of car A in between t = 0 to t = 8 sec. and motion of car B in between t = 0 tot = 7 sec., pick up the correct statement.

v (m/s)

10m/s

t=2s t=8st(s)

v (m/s)

10m/s

t=3s t=7st(s)

Car A Car B

(A) Distance travelled by car A is less than distance travelled by car B.

(B) Distance travelled by car A is greater than distance travelled by car B.

(C) Average speed of both cars are equal.

(D) Average speed of car A is less than average speed of car B.

50. For a particle moving along x-axis, which of the velocity versus position graphs given in options belowis/are possible : (position is represented by x-coordinate of the particle)

(A)

v

x (B)

v

x(C)

v

x(D) None of these

PHYSICS

E X E R C I S E 2PART - I : SUBJECTIVE QUESTIONS

1. A point traversed half the distance with a velocity v0. The remaining part of the distance was covered

with velocity v1 for half the time and with velocity v

2 for the other half of the time. Find the mean velocity

of the point averaged over the whole time of motion.

2. The displacement of a particle moving in a straight line is given by x = 16t – 2t2. Find out

(a) Displacement after 2 and 6 s. (b) Distance travelled after 2 and 6 s.

3. A man walking with a speed ' v

' constant in magnitude and direction passes under a lantern hanging at

a height H above the ground. Find the velocity with which the edge of the shadow of the man's headmoves over the ground, if his height is '

h

'.

4. A police jeep is chasing a culprit going on a moter bike. The motor bike crosses a turning at a speed of 72km/h. The jeep follows it at a speed of 90 km/h, crossing the turning ten seconds later than the bike.Assuming that they travel at constant speeds, how far from the turning will the jeep catch up with the bike?

5. A healthy youngman standing at a distance of 7 m from a 11.8 m high building sees a kid slipping from thetop floor. With what speed (assumed uniform) should he run to catch the kid at the arms height (1.8 m)?Take g = 9.8 m/s2.

6. An elevator is descending with uniform acceleration. To measure the acceleration , a person in the elevatordrops a coin at the moment the elavator starts. The coin is 6 ft above the floor of the elevator at time it isdropped. The person observes that the coin strikes the floor in 1 second. Calculate from these data theacceleration of the elevator. [Take g = 32 ft/s2]

7. A body starts with an initial velocity of 10 m/s and moves along a straight line with a constant acceleration.When the velocity of the particle is 50 m/s the acceleration is reversed in direction. Find the velocity ofthe particle when it reaches the starting point.

8. Two train are moving parallel to each other from station. First train with acceleration of0.5 m/s2 and get maximum velocity of 22 m/s. After 40 s. second train started with acceleration of 1 m/s2 and get a maximum velocity of 44 m/s. Find out the time after which, the second train catch the firsttrain.

9. A particle starts from rest traverses a distance ‘ d

’ with a uniform acceleration and then moves uniformly

with the acquired velocity over a further distance ‘ 2

d

’. Finally it comes to rest after moving through a

further distance ‘ 3

d

’ under uniform retardation. Assuming the entire path is a straight line, find the ratio

of the average speed over the journey to the maximum speed on the way.

10. The accompanying figure shows the velocity v of a particle moving on a coordinate line.

-4

2

(m/s)

(a) When does the particle move forward? move backward? Speed up? slow down?

(b) When is the particle's acceleration positive? Negative ? zero?

(c) When does the particle move at its greatest speed ?

(d) When does the particle stand still for more than an instant?

11. A particle starting from rest moves with constant acceleration . If it takes 5.0 s to reach the speed 18.0 km/h, find (a) the average velocity during this period,and (b) the distance travelled by the particle during thisperiod.

PHYSICS12. A particle moving with acceleration 4 m/s2 along x-axis covers 20 m in 4th second. Find the distance covered

by the particle in the 3rd and 5th seconds.

13. The two ends of a train moving with a constant acceleration pass a certain point with velocities u and

v. Show that the velocity with which the middle point of the train passes the same point is 2

vu 22.

14. A space craft flying in a straight course at 75 km/s fires its rocket motors for 6.0 s and then moves withthe constant speed attained. At the end of this time its speed is 120 km/s in the same direction.

(i) What was the space craft's average acceleration while the motors were firing?

(ii) How far did the space craft travel in the first 10 s after the rocket motors were started, themotors having been in action for only 6.0 s ?

15. A lift starts from the top of a mine shaft and descends with a constant speed of 10 m/s. 4 s later a boythrows a stone vertically upwards from the top of the shaft with a speed of 30 m/s. Find when and wherestone hits the lift. [ Take: g = 10 m/s² ]

16. For a particle moving along x-axis, v-t graph is shown below. OA, AB, BC, CD & DE are straight lines, but EF

is a curve having equation 210)(t

2

1v .

C

D

E G t(s)810 12

14

v(m/s)

Find,

(a) distance travelled and displacement of the particle in 6s.

(b) distance travelled and displacement of the particle from 6s to 10s.

(c) distance travelled and displacement of the particle from 10s to 14s.

(d) displacement of the particle from 0 to 14s.

(e) average speed and average velocity of the particle from 0 to 6s.

(f) average speed and average velocity of the particle from 0 to 10s.

(g) average speed and average velocity of the particle from 0 to 14s.

(h) acceleration of the particle at 1s, 3s, 6s, 9s and 12s.

(i) average acceleration of the particle from 0 to 4s, 0 to 10s, 0 to 14s, 2 to 8s and 4 to 14s.

17. A train stopping at two stations 2 kms apart on a straight line takes 4 minutes for the journey. Assuming

that its motion is first uniformly accelerated and then uniformly retarded. Prove thatx

1 + y

1 = 4, where

' x

' and '

y

' are the magnitude of the acceleration and retardation respectively.

18. On a 2lane road, car A is travelling with a speed of 36 km/h. Two cars B and C approach car A inopposite directions with a speed of 54 km/h each. At a certain instant, when the distance AB is equalto AC, both being 1 km, B decides to overtake A before C does. What minimum acceleration of car B isrequired to avoid an accident?

PHYSICSPART - II : OBJECTIVE QUESTIONS

19. Two balls of equal masses are thrown upwards, along the same vertical direction at an interval of 2seconds, with the same initial velocity of 40 m/s. Then these collide at a height of (Take g = 10 m/s2)

(A) 120 m (B) 75 m (C) 200 m (D) 45 m

20. A body starts from the origin and moves along the X-axis such that the velocity at any instant is givenby (4t3 – 2t), where t is in second and velocity in m/s. What is acceleration of the particle, when it is 2m from the origin.

(A) 28 m/s2 (B) 22 m/s2 (C) 12 m/s2 (D) 10 m/s2

21. The displacement of a body in motion is given by x = a sin (t + ). The time at which the displacementis maximum is ( & are constants)

(A)

(B)

2 (C) / 2 (D)

2

22. A body is released from the top of a tower of height h metre. It takes T seconds to reach the ground.Where is the ball at the time T/2 seconds ?

(A) at h/4 metre from the ground (B) at h/2 metre from the ground

(C) at 3h/4 metre from the ground (D) depend upon the mass of the ball

23. A stone is thrown vertically upward with an initial velocity u from the top of a tower, reaches the groundwith a velocity 3u. The height of the tower is:

(A) g

3u2

(B) g

4u2

(C) g

6u2

(D) g

9u2

24. A particle starts from rest with uniform acceleration a. Its velocity after n seconds is v. The displacementof the body in the last two seconds is :

(A) n

1)-2v(n(B)

n

1)-v(n(C)

n

1)v(n (D)

n

1)2v(2n

25. A ball is thrown vertically upwards from the top of a tower of height h with velocity v. The ball strikes theground after

(A)

2v

gh211

g

v(B)

2v

gh211

g

v(C)

2/1

2v

gh21

g

v

(D)

2/1

2v

gh21

g

v

26. A balloon is moving upwards with velocity 10 ms–1. It releases a stone which comes down to the groundin 11 s. The height of the balloon from the ground at the moment when the stone was dropped is :

(A) 495 m (B) 592 m (C) 460 m (D) 500 m

27. Water drops fall at regular intervals from a tap which is 5m above the ground. The third drop is leavingthe tap at the instant the first drop touches the ground. How far above the ground is the second drop atthat instant ? (Take g = 10 ms–2)

(A) 1.25 m (B) 4.00 m (C) 2.50 m (D) 3.75 m

28. Two particles held at different heights a and b above the ground are allowed to fall from rest. The ratioof their velocities on reaching the ground is :

(A) a : b (B) b:a (C) a2 : b2 (D) a3 : b3

PHYSICS29. In the one-dimensional motion of a particle, the relation between position x and time t is given by x2 +

2x = t Choose the correct statement :

(A) The retardation of the particle is 3)1x(4

1

(B) The uniform acceleration of the particle is 3)1x(

1

(C) The uniform velocity of the particle is 3)1x(

1

(D) The particle has a variable acceleration of 4t + 6.

30#. Consider the motion of the tip of the minute hand of a clock. In one hour

(A) the displacement is zero (B) the distance covered is zero

(C) the average speed is zero (D) the average velocity is zero

31#. Mark the correct statements for a particle going on a straight line:

(A) If the velocity and acceleration have opposite sign, the object is slowing down.

(B) If the position and velocity have opposite sign, the particle is moving towards the origin.

(C) If the velocity is zero at an instant, the acceleration should also be zero at that instant.

(D) If the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval.

32#. The velocity of a particle is zero at t = 0

(A) The acceleration at t = 0 must be zero

(B) The acceleration at t = 0 may be zero.

(C) If the acceleration is zero from t = 0 to t = 10 s, the speed is also zero in this interval.

(D) If the speed is zero from t = 0 to t = 10 s the acceleration is also zero in the interval.

33. Mark the correct statements :

(A) The magnitude of the instantanteous velocity of a particle is equal to its instantanteous speed.

(B) The magnitude of average velocity in an interval is equal to its average speed in that interval.

(C) It is possible to have a situation in which the speed of a particle is always zero but the average speed is

not zero

(D) It is possible to have a situation in which the speed of the particle is never zero but the average speed in

an interval is zero.

34. The velocity-time plot for a particle moving on a straight line is shown in fig.

10 20 30t(s)

-10

-20

10

0

v(m/s)

(A) The particle has constant acceleration

(B) The particle has never turned around.

(C) The particle has zero displacement

(D) The average speed in the interval 0 to 10s is the same as the average speed in the interval 10s to 20s.

PHYSICS35. Fig. shows the position of a particle moving on X-axis as function of time.

20

10

2 4 6 t(s)

x(m)

(A) The particle has come to rest 6 times

(B) The maximum speed is at t = 6 s

(C) The velocity remains positive for t = 0 to t = 6 s

(D) The average velocity for the total period shown is negative.

36. A body freely falling from rest has a velocity v after it falls through distance h. The distance it has to falldown further for its velocity to become double is :

(A) h (B) 2h (C) 3h (D) 4h

37. A person standing near the edge of the top of a building throws two balls A and B. The ball A is thrownvertically downward and the ball B is thrown vertically upward with the same speed. The ball A hits the groundwith a speed

A and the ball B hits the ground with a speed

B. Then we have :

(A) A >

B(B)

A <

B(C)

A =

B

(D) the relation between A and

B depends on height of the building above the ground

38. The velocity versus time graph of a body in a straight line is as follows :

Then the displacement of the body in 5 seconds is

(A) 3m (B) 5m (C) 4m (D) 2m

39#. A particle moves along the X-axis as

x = u(t – 2) + a(t – 2)2

(A) the initial velocity of the particle is u (B) the acceleration of the particle is a

(C) the acceleration of the particle is 2a (D) at t =2s particle is at the origin

40. A motor car is going due north at a speed of 50 km/h. It makes a 90º left turn without changing its speed. Thechange in the velocity of the car is about :

(A) 50 km/h towards west (B) 250 km/h towards south-west

(C) 250 km/h towards north-west (D) zero

41. Two objects moving along the same straight line are leaving point A with an acceleration a, 2 a &

velocity 2 u, u respectively at time t = 0. The distance moved by the object with respect to point A when

one object overtakes the other is :

(A) a

u6 2

(B) a

u2 2

(C) a

u4 2

(D) none of these

PHYSICS42. The displacement of a body from a reference point is given by, x = 2 t 3, where

' x ' is in metres and t in seconds. This shows that the body:

(A) is at rest at t = 3/2 (B) is accelerated

(C) is decelerated (D) is in uniform motion

43. A particle is thrown upwards from ground. It experiences a constant air resistance which can produce aretardation of 2 m/s2 opposite to the direction of velocity of particle. The ratio of time of ascent to the timeof descent is : [ g = 10 m/s2 ]

(A) 1 : 1 (B) 3

2(C)

3

2(D)

2

3

44. A particle moves along x-axis in positive direction. Its acceleration 'a' is givenas a = cx + d, where x denotes the x-coordinate of particle, c and d arepositive constants. For velocity-position graph of particle to be of type asshown in figure, the value of speed of particle at x = 0 should be.

(A) c

d4 2

(B) c

d2

(C) c

d2 2

(D) c

d8 2

45. Velocity (v) versus displacement (S) graph of a particle moving in a straight line is shown in figure.Corresponding acceleration (a) versus velocity (v) graph will be

(A)

10

10

(m/s)

(m/s2)a

(B)

10

10

(m/s)

(m/s2)a

(C)

10

(m/s)

(m/s2)a

(D)

10

10

(m/s)

(m/s2)a

46. A particle moves in x-y plane, starting from A , along straight line paths AB and then BC, as shown in thegraph. When it is at point P, angle between directions of its average velocity and instantaneous velocity is :

4

3

2

1

1 2 3 4x

y

(in m)

(in m)A

B

C

P(4,1)

45º

(A) 90º (B) 82º (C) 98º (D) 74º

PHYSICS47. A tiger running 100 m race, accelerates for one third time of the total time and then moves with uniform

speed. Then the total time taken by the tiger to run 100 m, if the acceleration of the tiger is 8m/s2, is :

(A) 53 s (B) 35 s (C) 12 s (D) 9 s

48#. A man in a lift ascending with an upward acceleration 'a' throws a ball vertically upwards with a velocity

‘v’ with respect to himself and catches it after ‘t1’ seconds. Afterwards when the lift is descending with

the same acceleration 'a' acting downwards the man again throws the ball vertically upwards with the

same velocity with respect to him and catches it after ‘t2’ seconds

(A) the acceleration of the ball w.r.t. ground is g when it is in air

(B) the velocity v of the ball relative to the lift is 21

21

tt

)tt(g

(C) the acceleration 'a' of the lift is 21

12

tt

)tt(g

(D) the velocity ‘v’ of the ball relative to the man is )tt(

ttg

21

21

PHYSICS

E X E R C I S E 3PART - I : MATCH THE COLUMN

1. Column gives some graphs for a particle moving along x-axis in positive x–direction. The variables v, x andt represent speed of particle, x–coordinate of particle and time respectively. Column gives certain resultinginterpretation. Match the graphs in Column with the statements in Column and indicate your answer bydarkening appropriate bubbles in the 4 × 4 matrix given in the OMR.

Column I Column II

(A) (p) acceleration is uniform and non zero

(B) (q) acceleration is non uniform

(C) (r) velocity is uniform

(D) (s) velocity is non-uniform

2. Column gives some graphs for a particle moving along x-axis in positive x–direction. The variables v, x andt represent speed of particle, x–coordinate of particle and time respectively. Column gives certain resultinginterpretation. Match the graphs in Column with the statements in Column and indicate your answer bydarkening appropriate bubbles in the 4 × 4 matrix given in the OMR.

Column Column

(A)

v

v - x graph

x(p) Acceleration of particle is uniform

PHYSICS

(B)

v2

v - x graph2

x

(q) Acceleration of particle is nonuniform

(C)

v

v - t graph

t(r) Acceleration of particle is directly proportional to ‘t’

(D)

v

v - t graph2

t2

(s) Acceleration of particle is directly proportional to ‘x’.

PART - II : COMPREHENSIONComprehension # 1

Read the following write up and answer the questions based on that.

The graph below gives the displacement of a particle travelling along the X-axis as a function of time. AM isthe tangent to the curve at the starting moment and BN is tangent at the end moment (1 = 2 =120°).

3. The average velocity during the first 20 seconds is

(A) – 10 m/s (B) 10 m/s (C) zero (D) 20 m/s

4. The average acceleration during the first 20 seconds is

(A) – 10 m/s2 (B) 10 m/s2 (C) zero (D) 20 m/s2

5. The direction ( i or – i ) of acceleration during the first 10 seconds is _____________ .

6. Time interval during which the motion is retarded.

(A) 0 to 20sec. (B) 10 to 20sec. (C) 0 to 10sec. (D) None of these

Comprehension # 2

45º

100m /s2 2

v (m /s )2 2 2

x(metres)A

A particle moves along x-axis. At time t = 0, the particle is at origin

and its initial velocity is along positive x-direction. The plot of its

(speed)2 versus its x–coordinate is as shown in graph.

PHYSICS7. As the particle moves the acceleration of particle :

(A) is always uniform (B) decreases with time

(C) increases with time (D) information insufficient

8. The time at which velocity of particle is zero.

(A) t = 5 sec. (B) t = 20 sec. (C) t = 25 sec. (D) t = 20 2 sec.

9. The velocity-time graph for the particle upto point A is best represented by (assuming velocity as positive inpositive x–direction.):

(A)

v(m/s)

t(sec)

(B)

v(m/s)

t(sec)

(C)

v(m/s)

t(sec)

(D)

v(m/s)

t(sec)

Comprehension # 3The plot of velocity versus time graph for a particle moving along a straight line is shown below. Answerthe following three questions based on given information.

t (sec.)t=3 sec.

=45°

v (m/s)

10. If v is velocity at any time t, then the value of dt

dv at t = 2 sec is :

(A) 1 m/s2 (B) – 1 m/s2 (C) 2 m/s2 (D) – 2 m/s2

11. The value of dot product of velocity and acceleration of particle at t = 2 sec. is :

(A) 1 m2/s3 (B) – 1 m2/s3 (C) 2 m2/s3 (D) – 2 m2/s3

12. If v is velocity at any time t, then the value of

5t

2t

dtv is (where v is in m/s and t is in seconds) :

(A) zero (B) 2.5 m (C) 1.5 m (D) 2 m

Comprehension # 4

The velocity 'v' of a particle moving along straight line is given in terms of time t as v = 3(t2 – t) where t is inseconds and v is in m/s.

13. The distance travelled by particle from t = 0 to t = 2 seconds is :

(A) 2 m (B) 3 m (C) 4 m (D) 6 m

14. The displacement of particle from t = 0 to t = 2 seconds is

(A) 1 m (B) 2 m (C) 3 m (D) 4 m

15. The speed is minimum after t = 0 second at instant of time

(A) 0.5 sec (B) 1 sec. (C) 2 sec. (D) None of these

Comprehension # 5

For a particle moving along x-axis, the acceleration a of the particle in terms of its x-coordinate x isgiven by a = – 9x, where x is in meters and a is in m/s2. Take acceleration, velocity and displacementin positive x-direction as positive. The initial velocity of particle at x = 0 is u = + 6 m/s.

16. The velocity of particle at x = 2 m will be :

(A) + 26 m/s (B) – 26 m/s (C) 72 m/s (D) 0

PHYSICS17. The maximum distance of particle from origin will be :

(A) 1 m (B) 2 m (C) 3 m (D) 4 m

18. The plot of velocity versus x-coordinate of particle in the duration it moves from origin towards thepositive x-direction, is best represented by :

(A)

v

x

(B)

v

x

(C)

v

x

(D)

v

x

PART - III : ASSERTION / REASON

19. Assertion : A particle having negative acceleration will slow down.

Reason : Direction of the acceleration is not dependent upon direction of the velocity.

(A) If both assertion and reason are true and reason is the correct explanation of assertion

(B) If both assertion and reason are true but reason is not the correct explanation of assertion.

(C) If assertion is true but reason is false

(D) If assertion is false but reason is true

20. Statement 1 : Magnitude of average velocity is equal to average speed.

Statement 2 : Magnitude of instantaneous velocity is equal to instantaneous speed.

(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True.

21. Statement 1 : When velocity of a particle is zero then acceleration of particle must be zero at thatinstant.

Statement 2 : Acceleration is equal to

dx

dvva , where v is the velocity at that instant..

(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True.

22. STATEMENT-1 : A particle moves in a straight line with constant accleration. The average velocity ofthis particle cannot be zero in any time interval

STATEMENT-2 : For a particle moving in straight line with constant acceleration, the average velocity

in a time interval is 2

vu , where u and v are initial and final velocity of the particle of the given time

interval.

(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

PHYSICS

E X E R C I S E 4JEE PROBLEMS (LAST 10 YEARS)

1. A particle of mass 102 kg is moving along the positive xaxis under the influence of a force

F(x) = 2x2

K where K = 102 N m2. At time t = 0 it is at x = 1.0 m and its velocity is v = 0. Find

(i) its velocity when it reaches x = 0.50 m(ii) the time at which it reaches x = 0.25 m. [ JEE '98, 8 ]

2. In 1.0 sec. a particle goes from point A to point B moving in a semicircle of radius 1.0 m. The magnitudeof average velocity is: [ JEE '99, 2 ]

(A) 3.14 m/sec (B) 2.0 m/sec (C) 1.0 m/sec (D) zero3. A ball is dropped vertically from a height d above the ground. It hits the ground and bounces up vertically

to a height d/2. Neglecting subsequent motion and air resistance, its velocity v varies with the height habove the ground as [ JEE '2000, 3 ]

(A) (B) (C) (D)

4. A block is moving down a smooth inclined plane starting from rest at time t = 0. Let Sn be the distance

travelled by the block in the interval t = n – 1 to t = n. The ratio 1n

n

S

S

is [JEE Scr. 2004, 3]

(A) n2

1n2 (B)

1n2

1n2

(C) 1n2

1n2

(D) 1n2

n2

5. A particle is initially at rest, It is subjected to a linear acceleration a , as shown in the figure. The maximum

speed attained by the particle is

[JEE Scr. 2004; 3]

(A) 605 m/s (B) 110 m/s (C) 55 m/s (D) 550 m/s6. The velocity displacement graph of a particle moving along a straight line is shown.

The most suitable acceleration-displacement graph will be [JEE Scr. 2005; 3]

(A) (B) (C) (D)

PHYSICS

E X E R C I S E 5AIEEE FLASH BACK

1. A car, moving with a speed of 50km/hr, can be stopped by brakes after at least 6m. If the same car is movingat a speed of 100km/hr, the minimum stopping distance is [AIEEE-2003](A) 12m (B) 18m (C) 24m (D) 6m

2. A ball is released from the top of a tower of height h meters. It takes T seconds to reach the ground. What is

the position of the ball at T

3 second. [AIEEE-2004]

(A) 8h

9 meters from the ground (B)

7h

9metres from the ground

(C) h

9meters from the ground (D)

17h

18meters from the ground

3. An automobile travelling with a speed of 60km/h, can brake to stop within a distance of 20m. If the car isgoing twice as fast i.e., 120km/hr, the stopping distance will be [AIEEE-2004](A) 60m (B) 40m (C) 20m (D) 80m

4. A car, starting from rest, accelerates at the rate f through a distance S, then continues at constant speed fortime t and then the decelerates at the rate f/2 to comes to rest. If the total distance traversed is 15S, then

[AIEEE-2005]

(A) 21

S ft6

(B) S = ft (C) 21

S ft4

(D) none of these

5. A particle is moving eastwards with a velocity of 5ms–1 . In 10 seconds the velocity changes of 5ms–1

northwards. The average acceleration in this time is [AIEEE-2005]

(A) 21ms

2 towards north (B) 21

ms2

towards north-east

(C) 21ms

2 towards north-west (D) zero

6. The relation between time t and distance x is t = ax2 + bx where a and b are constants. The acceleration is[AIEEE-2005]

(A) 2bv3 (B) –2abv2 (C) 2av2 (D) –2av3

7. A parachutist after bailing out falls 50m without friction. When parachute opens, it decelerates at 2m/s2. Hereaches the ground with a speed of 3m/s. At what height, did he bail out ? [AIEEE-2005]

(A) 182 m (B) 91m (C) 111m (D) 293m

8. A particle located at x = 0 at time t = 0, starts movign along with the positive x-direction with a velocity ‘v’ that

varies as v = x . The displacement of the particle varies with time as [AIEEE-2006]

(A) t2 (B) t (C) t1/2 (D) t3

9. The velocity of a particle is v = v0 +gt + ft2 . If its position is x = 0 at t = 0, then its displacement after unit time

(t = 1) is [AIEEE-2007]

(A) v0 + g/2 + f (B) v

0 + 2g + 3f (C) v

0 + g/2 + f/3 (D) v

0 + g + f

10. A spherical solid ball of volume V is made of a material of density 1. It is falling through a liquid of density

2

(2 <

1). Assume that the liquid applies a viscous force on the ball that is proportional to the square of its

speed v, i.e., Fviscous

= – kv2(k>0). The terminal speed of the ball is [AIEEE-2008]

(A)1Vg

k

(B)

1Vg

k

(C)

1 2Vg( )

k

(D)

1 2Vg( )

k

PHYSICS11. A body is at rest at x = 0. At t = 0, it starts moving in the positive x-direction with a constant acceleration.

At the same instant another body passes through x = 0 moving in the positive x-direction with a constantspeed. The position of the first body is given by x

1(t) after time ‘t’ and that of second body by x

2 (t) after

the same time interval. Which of the following graphs correctly describes (x1 – x

2) as a function of time

‘t’ ? [AIEEE-2008]

(A)

t

(x -x )1 2

O

(B) t

(x -x )1 2

O

(C)

t

(x -x )1 2

O

(D)

t

(x -x )1 2

O

12. A particle has an initial velocity of ˆ ˆ3i 4 j and an acceleration of ˆ ˆ0.4i 0.3j . Its speed after 10s is

[AIEEE-2009]

(A) 8.5 units (B) 10 units (C) 7 2 units (D) 7 units

13. Consider a rubber ball freely falling from a height h = 4.9 m onto a horizontal elastic plate. Assume that the

duraction of collision is negligible and the collision with the plate is totally elastic.Then the velocity as a

function of time and the height as a function of time will be [AIEEE-2009]

(A) t

t1 2t1 3t1 4t1

v

t

h

y

(B) t

v

t

h

y

O

v1

(C) t

v

t

h

y+v1

-v1

0

(D) t

v

t

h

y+v1

-v1

0 t1 2t1 3t1 4t1

PHYSICS

EXERCISE-11. Distance travelled by the car = 48 m,

Displacement of the car = 36 m

2. (a) 110 m (b) 50 m,tan–1

4

3 north to east

3. 3 km in north.

4. k23j4i19r

5. (B) 6. (D) 7. (A)

8. (a) 32 km/h (b) zero

9. (D) 10.313221

321

vvvvvv

vv3v

11. 50m, 5m/s, 25/9 m/s, 90 m

12. (B) 13. (A) 14. (A)

15. (D) 16. (B) 17. (C)

18. (a) [A] = [LT–1], [B] = [LT–3] ;

(b) 24 Bt ; (c) A + 300 B, 120 B

19. A – 3 Bt–4 20.2

5 = 2.5 m/s2

21. (A)

22. a = 3 m/s2 ; 2

175 = 87.5 m

23. (a) 2700 m = 2.7 km , (b) 60 m/s, (c) 225 m and2.25 km

24. 50m ; 2m/s2 ; 20 m

25. (a) 100 m ; (b) 10 s ; (c) 19 m, 15 m

26. 20m 27. (A) 28. (D)

29. (C) 30. (A) 31. (D)

32. (B) 33. (B) 34. (B)

35. (C) 36. (B) 37. (C)

38. distance travelled = 10 m; displacement = 6 m;

average velocity = 5

6 = 1.2 m/s

39.

1000 ft. , ,

ANSWER SHEET

straight line

parabolic curve

parabolic curve

40. v - t diagram for the two situations is shown below

tan = a1tan = 2a2

In v - t graph, distance travelled = area under the graph

41. (C) 42. (D) 43. (C)

44. (D) 45. (C) 46. (B)

47. (D) 48. (B) 49. (D)

50. (D)

EXERCISE-2

1.

210

21vvv2

vv0v2

2. (a) 24 m, 24 m (b) 24 m, 40 m

3.

hH

Hv 4. 1.0 km

5.10

49 m/s = 4.9 m/s

6. 20 ft/s2 7. 70 m/s

8. 62 s after the start of the second body

9. 5

3 = 0.6

10. (a) (0,1) s & (5,7)s

(1, 5)s

(1, 2) s & (5, 6) s

(0, 1) s & (3, 5) s & (6, 7) s

PHYSICS(b) (3, 6) s

(0, 2) s & (6, 7) s

(2, 3) s & (7, 9) s

(c) 0 s & (2, 3) s

(d) (7, 9) s

11. (a) 2.5 m/s , (b) 12.5 m

12. s3 = 16m, s

5 = 24m 13.v´ =

2

vu 22

14. (i) 7.5 km/s2 (ii) 1065 km

15. 12.9 s after the lift starts descending,

129 m below the top of the shaft

16. (a) 16m, 16m ; (b) 8m, 8m ; (c) 3

32m,

3

32m ;

(d) 3

56m (e)

3

8m/s,

3

8m/s ; (f)

5

12m/s,

5

4m/s

(g) 21

52m/s,

3

4 m/s, (h) 2, 0, –2, 2, 2 m/s2, (i)

1, 0, 7

4, –

3

4,

5

2m/s2

17. 4 18. 1 m/sec2 19. (B)

20. (B) 21. (B) 22. (C)

23. (B) 24. (A) 25. (A)

26. (A) 27. (D) 28. (B)

29. (A) 30#. (A) (D) 31#. (A) (B) (D)

32#. (B) (C) (D) 33. (A)

34. (A) (D) 35. (A) 36. (C)

37. (C) 38. (A) 39#. (C) (D)

40. (B) 41. (A) 42. (A) (B)

43. (B) 44. (B) 45. (A)

46. (B) 47. (A) 48#. (A) (C) (D)

EXERCISE-31. (A) p, s (B) r (C) p, s (D) q, s

2. (A) q, s (B) p (C) p (D) q, r

3. (A) 4. (C) 5. i

6. (C) 7. (A) 8. (B)

9. (D) 10. (A) 11. (B)

12. (C) 13. (B) 14. (B)

15. (B) 16. (D) 17. (B)

18. (D) 19. (D) 20. (D)

21. (D) 22. (D)

EXERCISE-4

1. (i) V

= 1 i m/s (ii) t = 4

3

3

2. (B) 3. (A) 4. (B)

5. (C) 6. (B)

EXERCISE-5

1. (C) 2. (A) 3. (D)

4. (D) 5. (C) 6. (D)

7. (D) 8. (A) 9. (C)

10. (D) 11. (A) 12. (C)

13. (D)