Question

2 skaters with the same mass face eachother standing still, both skaters push off of the other and they move in the opposite directions. Which of the following is true?
A.) They have zero momentum before pushing off

B.) The total momentum is 0 after pushing off

C.) They have equal and opposite momenta

D.) All of these statements are true

Please help, thanks!

Answer 1

D.) All of these statements are true

Explanation:

We can answer this question by using the law of conservation of momentum.

In fact, for an isolated system (=no external forces acting on it), the total momentum of the system must be conserved.

If we apply this law to this situation, we can say that the total momentum of the 2 skaters before pushing off must be equal to the total momentum of the 2 skaters after pushing off.

Before pushing off, the total momentum of the 2 skaters is zero:

$p_i =0$

Because they are both still, so the total momentum is

$p_i = m_1 u_1 + m_2 u_2 = 0$

Since $u_1 = u_2 = 0$ (where $u_1, u_2$ are the initial velocities of the 2 skaters, while $m_1,m_2$ are their masses)

So, the statement

A.) They have zero momentum before pushing off

is true.

So, according to the law of conservation of momentum, the final momentum must be equal to the initial total momentum:

$p_f = p_i$

And so

$p_f = 0$

Therefore, the statement

B.) The total momentum is 0 after pushing off

is also true.

Finally, the total final momentum can be written as

$p_f = p_1 + p_2$

where

$p_1$ is the momentum of the first skater after pushing off

$p_2$ is the momentum of the second skater after pushing off

Since $p_f = 0$, this implies that

$p_1+p_2 = 0\\p_2 = -p_1$

So, the statement

C.) They have equal and opposite momenta

is also true.

Therefore the correct option is

D.) All of these statements are true

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