# Two balls of different mass elastically collide

## Homework Statement

A tennis ball of mass m_t is held just above a basketball of mass m_b. With their centers vertically aligned, both are released from rest at the same moment, so that the bottom of the basketball falls freely through a height h and strikes the floor. Assume an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down because the balls have a very small amount of separation while falling.

(Q1.) The two balls meet in an elastic collision. To what height does the tennis ball & basketball rebound?
(Q2.) How do you account for the height being larger than h? Does that seem like a violation of conservation of energy?

## Homework Equations

(1.) v_1f (basket ball) = [(m_1-m_2)/(m_1+m_2)]v_1i + [(2*m_2)/(m_1+m_2)]v_2i

(2.) V_2f (tennis ball) = [(2*m_1)/(m_1+m_2)]v_i + [(m_2-m_1)/(m_1+m_2)]

(3.) v_1i (basketball) = -v_2i (tennis ball) →
v_1i = + √(2gh)
v_2i = -√(2gh)

*Note: Recall, m_basketball ≈ 11*m_tennis ball

## The Attempt at a Solution

*I already have the correct solution and answer to the first questions but I don't understand it. I need help with understanding the story behind the mathwork.

-Solution 1a:
Answer: Tennis ball will rise up to ~7 times of its original height approximately.

-Solution 1b:
(i) v_1f (basket ball) = [(m_1-m_2)/(m_1+m_2)](√(2gh)) + [(2*m_2)/(m_1+m_2)](-√(2gh))
(ii) =[(m_b-3*m_t)/(m_b+m_t)](√(2gh))
(iii) =(8/12)(√(2gh))

****how does this (direclty below, iv) relate to the last part of my above solution (iii):
(iv) (1/2)(m_b*v^2_1f)=m_b*g*h_1f)
(v) h_1f = (v^2_1f)/(2*g) = h*(2/3)^2 ≈ 0.44*h
Answer: Basketball will rise up to ~.44 times of its original height approximately.

My Questions:
A. I can see that the equations given:
(1.) v_1f (basket ball) = [(m_1-m_2)/(m_1+m_2)]v_1i + [(2*m_2)/(m_1+m_2)]v_2i
(2.) V_2f (tennis ball) = [(2*m_1)/(m_1+m_2)]v_i + [(m_2-m_1)/(m_1+m_2)]

relates to the conservation of momentum (which I'm guessing will be the equation that I am supposed to always think of first when it comes to elastic collisions) but how and why did we subtract mass in the equations???? instead of just using m_1*v_1i + m_2*v_2i = m_1*v_1f + m_2*v_2f

Note: I am really struggling with Physics, helping me understand the material and learning it...I would really appreciate it. I sit at one problem for hours, which eats up my time in studying my other course material.

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