I am trying to figure out why they said that the right answer is 2?
The left figure depicts the path of two colliding steel balls A and B. Which of the arrows labeled from 1 to 5 represents teh direction of the impulse exerted on ball B by ball A?
Shouldn't it be 5? because the impulse exerted on B,$~~\vec{I}$, is the variation of the momentum $~~\vec{p}$,
$$\vec{I} = \vec{p_{f}}-\vec{p_{i}} = m (\vec{v_{f}}-{\vec{v_{i}}}) $$
that means that the impulse has the same direction of the vector $~\vec{v_{f}}-{\vec{v_{i}}}$ but as we can see from the picture, $~\vec{v_{f}}-{\vec{v_{i}}}$ is corresponding to the arrow 5, so the impulse too has to have the same direction, isn't it ?
Best Answer
The change in momentum of B is given by,
$I = p_f - p_i = m(v_f - v_i)$
Now $v_f$ and $v_i$ each have a vertical and a horizontal component.
The horizontal component of $v_f$ is equal to the horizontal component of $v_i$ (both pointing to the right).
Therefore, the difference in moment is due to only the difference in the vertical component.
The initial vertical component $v_{i_y}$ is equal and opposite to the final vertical component, $v_{f_y}$.
Therefore,
$I = m(v_{f_y} - v_{i_y}) = m(-v_{i_y} - v_{i_y}) = m(-2v_{i_y})$
and thus the impulse is in the opposite direction to the initial velocity.