Question: Observe the behavior of Newton's Cradle. Momentum conservation is satisfied when one ball is pulled back and released, resulting in a collision such that one ball on the other end moves away at the same speed. Suppose each ball has mass, m, and the velocity of each released ball just before the collision is v. By determining the total momentum and kinetic energy before and after the collision, show that momentum and kinetic energy are conserved in this case (consider the instances in time just before and just after the collision, and neglect and collisions occurring between the stationary balls.) Momentum would also be conserved if 2 balls moved away at half the initial pre-collision speed. Show that this would be true by determining the total momentum before and after the collision. Would the kinetic energy be conserved in this second scenario? Determine the initial and final kinetic energy to find out.

OneClass: Question: Observe the behavior of Newton's Cradle. Momentum conservation is satisfied when ...

A playground ride consists of a disk of mass M = 56 kg and radius R= 2.0 m mounted on a low-friction axle. A child of mass m = 21 kgruns at speed v = 2.8 m/s on a line tangential to the disk andjumps onto the outer edge of the disk.

ANGULAR MOMENTUM
(a) Consider the system consisting of the child and the disk, butnot including the axle. Which of the following statements are true,from just before to just after the collision?
The momentum of the system doesn't change.
The momentum of the system changes.
The axle exerts a force on the system but nearly zero torque.
The torque exerted by the axle is zero because the force exerted bythe axle is very small.
The angular momentum of the system about the axle changes.
The angular momentum of the system about the axle hardlychanges.
The torque exerted by the axle is nearly zero even though the forceis large, because || is nearly zero.

(b) Relative to the axle, what was the magnitude of the angularmomentum of the child before the collision?
|C| = kgÂ·m2/s
(c) Relative to the axle, what was the angular momentum of thesystem of child plus disk just after the collision?
|C| = kgÂ·m2/s
(d) If the disk was initially at rest, now how fast is it rotating?That is, what is its angular speed? (The moment of inertia of auniform disk is Â½MR2.)
= radians/s
(e) How long does it take for the disk to go around once?
Time to go around once = s
ENERGY
(f) If you were to do a lot of algebra to calculate the kineticenergies before and after the collision, you would find that thetotal kinetic energy just after the collision is less than thetotal kinetic energy just before the collision. Where has most ofthis energy gone?
Increased chemical energy in the child.
Increased thermal energy of the disk and child.
Increased translational kinetic energy of the disk.

MOMENTUM
(g) What was the speed of the child just after the collision?
v = m/s
(h) What was the speed of the center of mass of the disk just afterthe collision?
vcm = m/s
(i) What was the magnitude of the linear momentum of the disk justafter the collision?
|p| = kgÂ·m/s
(j) Calculate the change in linear momentum of the systemconsisting of the child plus the disk (but not including the axle),from just before to just after impact, due to the impulse appliedby the axle. Take the x axis to be in the direction of the initialvelocity of the child.
px = px,f - px,i = kgÂ·m/s

ANGULAR MOMENTUM
(k) The child on the disk walks inward on the disk and ends upstanding at a new location a distance R/2 = 1 m from the axle. Nowwhat is the angular speed? (It helps to do this analysisalgebraically and plug in numbers at the end.)
= radians/s
ENERGY
(l) If you were to do a lot of algebra to calculate the kineticenergies in part (k), you would find that the total kinetic energyafter the move is greater than the total kinetic energy before themove. Where has this energy come from?
Decreased chemical energy in the child.
Decreased momentum of the disk.
Decreased thermal energy of the disk and child.

As shown in the figure (Figure 1) , a superball with mass m equal to 50 grams is dropped from a height of hi =1.5 m. It collides with a table, then bounces up to a height of hf = 1.0 m. The duration of the collision (the time during which the superball is in contact with the table) is tc = 15ms. In this problem, take the positive y direction to be upward, and use g = 9.8 m/s2 for the magnitude of the acceleration due to gravity. Neglect air resistance.

a. Find the y component of the momentum, p_before,y, of the ball immediately before the collision.

b. Find the y component of the momentum of the ball immediately after the collision, that is, just as it is leaving the table.

c. Find Jy, the y component of the impulse imparted to the ball during the collision.

d. Find the y component of the time-averaged force Favg,y, in newtons, that the table exerts on the ball.

e. Find K_after - K_before, the change in the kinetic energy of the ball during the collision, in joules.

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