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1D Kinematics & Vectors

Look at the new items on the webpage Attend your tutorial section

tomorrow. Opened 2 more seats in room

G2B60 for tutorials – can see Leigh Dodd if need help

Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/

Announcements:

•  Your 4 digit CAPA number changes every week. Use the correct one.

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Summary of constant acceleration equations

No displacement in this equation

No final velocity in this equation

No time in this equation

No acceleration in this equation

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A.  more than 9.8 m/s2 B.  9.8 m/s2 C.  less than 9.8 m/s2 D.  depends on the mass of the ball

Clicker question 1 Q. If you drop an object in the absence of air resistance, it accelerates downward at 9.8 m/s2. If you throw it downward, its downward acceleration after release is

Set frequency to BA

The initial velocity has nothing to do with it! After release, the only acceleration is due to gravity which is 9.8 m/s2

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Clicker question 2 Set frequency to BA

Q. A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit the ground below the cliff with the greater speed is the one initially thrown

A: upward. B: downward. C: neither—they both hit at the same speed

If you throw a ball up with v0, it will go up, stop, come back down... and when it reaches you, it will have the SAME speed as it started with, just going the other way. That means that either ball has the same (downward) speed when it goes by you - the one you threw up just takes longer...

If you prefer a formula: vf2 = v0

2 + 2*a*(height). In this case, v0 is either + or -, but when you square it, you get the same result! vf (final speed) is going to be the same either way.

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Example 1: Deer in the Headlights The driver of a car going 30 m/s on a straight road sees a deer in the road 75 m ahead. What acceleration is needed to stop before hitting the deer?

What do we know? Solution 1: Using we get acceleration:

First, draw a picture 30 m/s 75 m

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Example problem 1 (continued) What do we know?

Solution 2: From we get

Then solve to get

30 m/s 75 m

Acceleration is negative because we defined velocity as positive and car is slowing

Sometimes give acceleration in g’s:

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A.  –5 m/s2 B.  –10 m/s2 C.  –15 m/s2 D.  –20 m/s2 E.  5 m/s2

Clicker question 3 Q. On planet X, a cannon ball is fired straight upward. The position and velocity of the ball at many times are listed at the right. Note that we have chosen up as the positive direction. What is the acceleration due to gravity on Planet X?

Set frequency to BA Time (s)

Height (m)

Velocity (m/s)

0 0 20 1 17.5 15 2 30 10 3 37.5 5 4 40 0 5 37.5 -5 6 30 -10 7 17.5 -15 8 0 -20

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Solutions to clicker question 3 Time (s)

Height (m)

Velocity (m/s)

0 0 20 1 17.5 15 2 30 10 3 37.5 5 4 40 0 5 37.5 -5 6 30 -10 7 17.5 -15 8 0 -20

Solution 1: Use for constant acceleration to get:

Solution 2: Use and solve for a:

Solution 3: Use with x=40 m, x0=0 m, v0=20 m/s, t = 4 s to get

Solution 4: Use with x=0 m, x0=40 m, v0=0 m/s, t = 4 s to get so

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Clicker question 4 Set frequency to BA A truck traveling at 50 km/hr (about 14 m/s) approaches a car stopped at a red light. When the truck is 100 m from the car, the light turns green and the car immediately begins to accelerate at 2.0 m/s2 to a final speed of 100 km/hr. Which graph represents this situation?

car t

x

truck

A car

t

x

truck

B car

t

x

truck

C D. None of the graphs

Can't be A or B, because in both of those the truck (straight line, i.e. constant v) is at or AHEAD of the car at t=0, where the light turned green. But in the problem, the truck is 100 m behind the car...So it's either C or D. The question is... does the truck ever pass the car (do the curves cross) or not? Need to do this with math!... Consider the time when the car has first reached the same speed as the truck... When/where does that happen?

For constant a, Δv= a*t, so the car reaches 14 m/s at t=Δv/a = (14 m/s -0) /(2 m/s^2 )= 7 seconds. At that time, x(car) = 1/2*a*t^2 = (1/2)*(2 m/s^2)*(7 sec)^2 = 49 m. At that same time, Δx(truck) = 14 m/s * 7 sec = 98 m. Since it started off 100 meters back, the truck is STILL behind the car at the instant they have the same speed. That means curve C is reasonable!

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Clicker question 5 Set frequency to BA

A ball is launched straight up with initial velocity vo. (Neglect air resistance) If the initial velocity vo is doubled, the time to reach the apex of the trajectory...

A: doubles. B: increases by a factor of 4. C: Neither of these. D: Not enough information given.

Constant accel says vf = v0 + a*time, and here, vf = 0 because we're at the top. So, v0 = -a * time, which means if you double v0, you double the time.

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Clicker question 6 Set frequency to BA

A ball is launched straight up with initial velocity vo. (Neglect air resistance) If the initial velocity vo is doubled, the maximum height of the ball….

A: doubles.

B: increases by a factor of 4.

C: Neither of these.

D: Not enough information given.

Constant accel says vf2 = v0

2 + 2*a*height. Again, with vf = 0, we have v0

2 = - 2*a*height. So, if you double v0, that "squared" means you quadruple the height!

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2D & 3D Kinematics •  Displacement, velocity, & acceleration are vectors and

so have magnitude and direction (which is why we needed to remember signs in 1D)

•  We will mostly work in 2D for the next couple of weeks but everything generalizes to 3D

Trigonometry refresher: SOHCAHTOA

Adjacent

Opposite Hypotenuse

Need to understand trig functions (including inverse) on calculator and understand degrees and radians

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Vector representations •  Can represent as an arrow •  Can represent as ordered list of

numbers in a known coordinate system (giving the components)

•  Can represent as a sum of components times unit vectors – Unit vectors have magnitude 1

and point along an axis:

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Vector representation y

x 4

3 Vector is represented by arrow

The vector or A is written with an arrow or in bold The magnitude is written in normal font |A| Components are which, if necessary, can be found by trigonometry:

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Graphical vector addition Want to find where and

Rule is head to tail:

Vector addition is commutative:

Vector subtraction done by adding the negative:

Can multiply vector by scalar multiplying the length but leaving the direction unchanged except a negative scalar flips direction 180°

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Clicker question 7 Set frequency to BA

Q. Three vectors, are shown. Which answer represents the vector ?

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

A

B C

(E) None of these.

A B

C

S

So

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Vector addition by components Split vectors into orthogonal components and add components individually. For and

then

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Vector addition by components (2) Use trigonometry to split vectors into orthogonal components if you are given magnitude and direction

but angles can be tricky

Draw quick diagram and determine angle

-6.5

3.2

Need to determine how measured angle relates to angle from +x axis. In this case

€

C = (7.2,154°)

Last Slide •  Don’t forget Tutorial material for your recititation

tomorrow!

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