Advanced Problems 1

These problems will contain:1. Basic concepts of velocity and

acceleration.2. Usage of proper physics 1

kinematics equations.3. Some calculus techniques,

such as derivatives and integration.

1. An indestructible bullet 2cm long is fired straight through a board that is 10cm thick. The bullet strikes the board with a speed of 420m/s and emerges with a speed of 280m/s.

– What is the average acceleration of the bullet as it passes through the board?

– What is the total time that the bullet is in contact with the board?

– What thickness of board would it take to stop the bullet (assuming the bullet’s acceleration though the board remains the same)

2. A student throws a set of keys vertically upward to her sorority sister, who is in a window 4m above. The keys are caught 1.5s later by the sisters outstretched hand.

– With what initial velocity were the keys thrown?

• What was the velocity of the keys just before they were caught?

3. The height of a helicopter above the ground is given by h = 3t3, where h is in meters and t is in seconds. After 2 seconds, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground?

4. Automotive engineers refer to the time rate of change of acceleration as the “jerk”. If an object moves in one dimension such that its jerk J is constant,

a. Determine expressions for its acceleration, velocity, and position given that its initial acceleration, velocity and speed are ai, vi, and

xi respectively.

b. Show that )(222ii vvJaa

5. The speed of a bullet as it travels down a barrel of a rifle toward the opening is given by v = -5x107t2 + 3x105t. The acceleration of the bullet as it just leaves the barrel is zero.

a. Determine the acceleration and position of the bullet as a function of time when the bullet is in the barrel.

b. Determine the length of time the bullet is accelerated.

c. Find the speed at which the bullet leaves the barrel.

d. What is the length of the barrel?

6.A daring ranch hand sitting on a tree limb wishes to drop vertically onto a horse galloping under a tree. The speed of the horse is 10m/s, and the distance from the limb to saddle is 3m.

What must be the horizontal distance between the limb and the saddle when the ranch hand drops?

7.A test rocket is fired vertically upward from a well. A catapult gives it an initial velocity of 80m/s at ground level. Subsequently, its engines fire and it accelerates upward at 4m/s2 until it reaches an altitude of 1000m. At that point the engines fail, and the rocket goes into free fall.

a.How long is the rocket in motion above the ground?

b.What is its maximum altitude?

c. What is its velocity just before it hits the ground?

8.A physics student and a mountain climber climbs a 50m cliff that overhangs a calm pool of water. He throws two stones vertically downward, 1second apart, and observes that they cause a single splash. The first stone has an initial speed of 2m/s.

• How long after the release of the first stone do the stones hit the water?

• What was the initial velocity of the second stone?

• What is the velocity of each stone at the instant the two hit the water?

9.You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the x axis and at a constant height of 7.6x103m. At t=0, the plane is directly above you. At t=30s, the position vector from you to it is given by P30=(8.04x103m)i + (7.6x103m)j. Determine the magnitude and orientation of the plane’s position at t=45s.

10.Given the displacement vectors

A= (3i-4j+4k)m and B= (2i+3j-7k)m find the magnitudes of C = A + B and D = 2A – B also expressing each in terms of its x, y, and z components.

11.A radar station locates a sinking ship at range 17.3km and bearing 136° clockwise from north. From the same station a rescue plane is at horizontal range 19.6km 153° clockwise from north. With elevation 2.2 km.

a.Write the vector displacement from plane to ship letting i represent east, j represent north, and k represent up.

b.How far apart are the plane and ship?

12.A vector is given by R=2i + 1j +3k.

a.Find the magnitudes of the x, y, and z components.

b. Find the magnitude of R.

c. Find the angles between R and the x, y, and z axes.

13. In general, the instantaneous position of an object is specified by its position P leading from a fixed origin to the location of the object. Suppose that P= 4i+3j-2tj where P is in meters and t is in seconds what is the derivative of this position function? What does this derivative represent about the object?

14. A jet airliner, moving initially at 300mi/h to the east, suddenly enters a region where the wind is blowing at 100mi/h in a direction 30° north of east. What are the new speed and direction of the aircraft relative to the ground?

15. A fish swimming in a horizontal plane has velocity vi = (4i + 1j)m/s at a point in the ocean whose displacement from a certain rock is ri = (10i – 4j)m. After the fish swims with constant acceleration for 20s, its velocity is v = (20i + 5j)m/s

a. What are the components of the acceleration?

b. What is the direction of the acceleration with respect to the unit vector i?

c. Where is the fish at t=25s if it maintains its original acceleration and in what direction is it moving?

16. A particle initially located at the origin has an acceleration of a = 3j m/s2 and an initial velocity of vi = 5i m/s.

a. Find the vector position and velocity at any time t.

b. Find the coordinates and speed of the particle at t=2s.

17. A projectile is fired in such a way that its horizontal range is equal to 3 times its maximum height. What is the angle of projection?

18. A placekicker must kick a football from a point 36m from the goal, and half the crowd hopes it will clear the crossbar, which is 3.05m high. When kicked, the ball leaves the ground with a speed of 20m/s at an angle of 53° to the horizontal.

a. By how much does the ball clear or fall short of the crossbar?

b. Does the ball approach the crossbar while still rising or while falling?

19. A train slows down as it rounds a sharp horizontal curve, slowing from 90km/h to 50km/h in the 15 seconds that it takes to round the curve. The radius of the curve is 150m. Compute the acceleration at the moment the train speed reaches 50km/h. Assume that the train slows down at a uniform rate during the 15 second interval.

20. The determined coyote is out once more to capture the roadrunner. The coyote wears Acme jet powered roller skates which provide constant horizontal acceleration of 15m/s2. The coyote starts off at rest 70m from the edge of a cliff the instant that the roadrunner zips past him in the direction of the cliff.

a. If the roadrunner moves with constant speed, determine the minimum speed he must have to reach the cliff before the coyote.

b. If the cliff is 100m above the floor of the canyon, determine where the coyote lands after missing the roadrunner (his skates still thrust while in midair).

c. What is his impact velocity?