Physics question?

Question

2 people stand on 2.0m platform, one on each end. Platform floats parallel to the ground on air, like hovercraft. One person throws 6.0kg ball to the other, who catches it. The ball travels nearly horizontally. Excluding the ball, the total mass of the platform and people is 118kg. Because of the throw, this 199kg mass recoils. How far does it move before coming to rest again?

*I know this question has something to do with the conservation of momentum but I just don't get how it can be solved without knowing the velocities of any member of the system. Also, if you can show your work in solving this question, that would be great. Thanks a lot!

Answer 1

Yes it has to do with conservation of momentum. Moreover it has to do with relative motion.

From conservation of momentum you find how the two velocities a related to each other.
Because ball (_b) and platform (_p) were initially at rest the balance of momentum before the throw and a moment during the flight of the ball is:
0 = p_p + p_b
<=>
0 = m_p·v_p + m_b·v_b
=>
v_p = - m_b/m_p · v_b
where v are the velocities with respect to the ground.

Next step is to find the time of flight.
The ball moves relative to the platform with a relative velocity of:
v_r = v_b - v_p = (1 + m_b/m_p) · v_b
which is constant. Hence the distance the ball travels along the platform is:
Δs_r = v_r · Δt
The time elapsed until the ball reaches the other end of the platform at Δs_r' = 2m is
Δt' = Δs_r' / v_r = Δs_r' /{(1 + m_b/m_p) · v_b}

During this period the platform moves relative to the ground with v_p. The distance is
Δs_p' = v_p · Δt'
= - m_b/m_p · v_b · Δs_r' /{(1 + m_b/m_p) · v_b}
= - (m_b/m_p)/(1 + m_b/m_p) · Δs_r'
= - (6/118)/(1 + 6/118) · 2.0m
= - 0.0968m
(negative sign indicates that the platform moves in opposite direction to the ball)

Answer 2

Simply put: From conservation of momentum, the CG of the total system doesn't move. Let W be the platform width, M1 & X1 the platform + people mass & CG, M2 & X2 the ball mass & CG, and (i) = initial, (f)=final.
Then
M1X1(i) + M2X2(i) = M1X1(f) + M2X2(f)
M1(X1(f) - X1(i)) = M2(X2(i) - X2(f)) = M2 * W
X1(f) - X1(i) = W * M2 / M1 (answer)

Answer 3

What 199kg mass? Is that a typo?

And yes, use conservation of momentum.

p platform = pball

v platform = vball mball / mplatform

vball*time = width of platform, so vball = width/time

distance platform moves = v platform * time

= width * mball / mplatform --time cancels