AP Physics C I.D

Systems of Particles and Linear Momentum

“The alteration of motion is proportional to the force

impressed” – Isaac Newton (as related in a private

conversation to W. Evans)

What Newton means by “motion”p = mv

Momentum and Newton’s Second Law Revisited

Ex. A golfer strikes a ball with a mass of 0.050 kg. The time of impact between the golf club and the ball is 1.0 ms. If the ball leaves the face of the club with a speed of 70.0 m/s, calculate the average force exerted on the ball by the club.

Examples of Impulse

Impulse and F vs. t graphsF

t

t1 t2

Ex. One legged man punts a football (mass 0.40 kg) and gives it a launch speed of 30.0 m/s. Find the impulse delivered to the football by the man’s foot and average force exerted by the kicker on the ball, if the impact time is 8.0 ms.

Ex. An 80.0 kg stuntman jumps out of a window that is 45 m above ground. a) How fast is he falling when he reaches ground level? b) He lands on a large air mattress, coming to rest in 1.5 s. What average force does he experience while coming to rest? c) What force would he experience if he landed on the ground (impact time 10.0 ms) instead of the air mattress?

Ex. A small block of mass 0.070 kg, initially at rest, experiences a force F for 10.0 ms. The force varies with time as shown by the graph below. What is the final speed of the block?

F (N)

t (ms)

20

1062

Conservation of Linear Momentum (Collisions and

Explosions)

Ex. An astronaut is floating in space near her shuttle when she realizes that the cord that attaches her to the ship has become disconnected. Her total mass (body, suit and equipment) is 91 kg. She reaches into her pocket, finds a 1.0 kg metal tool, and throws it directly away from the ship with a speed of 9.0 m/s. If the ship is 10.0 m away, how long will it take her to reach it?

Rifle Recoil: Find the recoil velocity of a 5.0 kg rifle that shoots a 0.050 kg bullet at a speed of 120 m/s.

Signs of a momentum problem

• A collision occurs

• A single object “explodes” into parts

• An internal force (not external) causes relative motion between two objects

Collisions

• Elastic: momentum is conserved, kinetic energy is conserved

• Inelastic: momentum is conserved (if no external forces act on the system), kinetic energy is not conserved

Ex. Two balls roll toward each other. The red ball has mass of 0.50 kg and a speed of 4.0 m/s just before impact. The green ball has a mass of 0.20 kg and a speed of 2.0 m/s. After the head-on collision, the red ball continues in its initial direction with a speed of 2.0 m/s. Find the velocity of the green ball after the collision. Is the collision elastic or inelastic?

Ex. Two blobs of clay are thrown towards each other and collide in mid-air. The red clay has a mass of 0.50 kg and a speed of 4.0 m/s just before impact. The green clay has a mass of 0.30 kg and a speed of 2.0 m/s. If the collision is completely inelastic, determine the velocity of the composite object immediately after the collision.

Ex. Ballistics Pendulum. A bullet with mass of 9.5 g strikes a 5.4 kg wooden block of a ballistics pendulum. The embedded bullet and block rise to a height of 6.3 cm. What is the initial velocity of the bullet?

Collision in two dimensions

Ex. A 500 kg car travels at 20.0 m/s due north. It collides with a 500 kg traveling due west at 30.0 m/s. The cars lock bumpers and stick together. What is the velocity of the combined cars immediately after impact?

Ex. An object of mass m moves with velocity v toward stationary object of mass 2m. After impact, the objects move off in directions as shown. a) Determine the magnitudes of the velocities after the collisions in terms of v. b) Is this an elastic or inelastic collision?

“I tossed a baseball (and a bat for good measure) into the

air . . . .”

Center of Mass – a) the point on an object where all the mass

can be considered to be concentrated b) all external

forces are applied there

So, for the Second Law, the net force causes the center of

mass to accelerate.

Center of mass for a system of discrete objects

Ex. Two objects, one of mass m and one of mass 2m, hang from light threads from the ends of a uniform bar of length 3L and mass 3m. The masses m and 2m are at distances L and 2L, respectively, shown below the bar. Find the center of mass of the system.

m

3m

2m

L

2L

3L

Ex. (This type of problem has been on a couple of AP MC exams) A man of mass m is standing at one of a floating stationary barge of mass 3m. He then walks to the other end of the barge, a distance of L meters. Ignore frictional effects between the barge and the water. a) How far will the barge move? b) If the man walks with an average velocity vm, what is the velocity of the barge vb?

Time out! A note on density

For an entire object For a mass element