Two balls of equal mass are thrown horizontally with the same initial velocity. they hit identical stationary boxes resting on a frictionless horizontal surface. the ball hitting box 1 bounces back, while the ball hitting box 2 gets stuck. which box ends up moving faster? a. box 1b. box 2c. same

a. Box 1

Explanation:

Hi there!

The momentum of the system box-ball is conserved in both cases because there is no external force applied on the system.

The momentum of the system is calculated as the sum of the momenta of each object that composes the system. The momentum is calculated as follows:

p = m · v

Where:

p =  momentum.

m = mass.

v = velocity.

Then, the momentum of the system before and after the collision will be:

System ball - box 1

initial momentum = final momentum

mb · vb + m1 · v1 = mb · vb´ + m1 · v1´

Where:

mb = mass of the ball.

vb = veloctiy of the ball.

m1 = mass of box 1.

v1 = velocity of box 1.

vb´ = final velocity of the ball.

v1´ = final velocity of box 1.  

Since the initial velocity of the box is zero:

mb · vb = mb · vb´ + m1 · v1´

Solving for v1´

mb · vb - mb · vb´ = m1 · v1´

mb · (vb - vb´) = m1 · v1´

mb · (vb - vb´) / m1 = v1´

Since vb´ is negative because the ball bounces back, then:

mb · (vb + vb´) / m1 = v1´

Now let´s express the momentum of the system ball - box 2

System ball -box 2

mb · vb + m2 · v2 = (mb + m2) · v2´

Since v2 = 0

mb · vb =  (mb + m2) · v2´

Solving for v2´:

mb · vb / (mb + m2) = v2´

Comparing the two expressions:

v2´ = mb · vb / (mb + m)

v1´ = mb · (vb + vb´) / m

In v1´ the numerator is greater than the numerator in v2´ because

vb + vb´> vb  

In v2´ the denominator is greater than the denominator in v1´ because

mb + m > m

then v1´ > v2´

Box 1 ends up moving faster than box 2