Collisions in Two Dimensions — Question 9

Question 9

Two upward departures contradict zero transverse input

A $1\,\mathrm{kg}$ projectile moving along $+x$ hits a stationary $2\,\mathrm{kg}$ target. Both final speeds are strictly positive. A report places both outgoing directions between $10^\circ$ and $30^\circ$ above $+x$. The system is said to receive no external impulse.

1. Test whether any speeds can satisfy momentum conservation.

2. If the speeds are measured as $3$ and $2\,\mathrm{m/s}$, bound the necessary external $y$ impulse.

Question 9 - Solution

Two upward departures contradict zero transverse input

(a) Derivation and calculation

The initial transverse momentum vanishes, but the reported final value is $$P_{fy}=m_1v_1\sin\theta_1+m_2v_2\sin\theta_2.$$ With positive masses and speeds and $10^\circ\leq\theta_i\leq30^\circ$, $$P_{fy}>0=P_{iy}\Rightarrow\boxed{\text{zero external impulse is impossible}}.$$

(b) Result and interpretation

For the measured speeds, $$J_y=3\sin\theta_1+4\sin\theta_2.$$ Sine is increasing throughout the allowed angular range, so $$\boxed{7\sin10^\circ\leq J_y\leq7\sin30^\circ},\qquad\boxed{1.216\leq J_y\leq3.50\ \mathrm{N\,s}}.$$

Physical check

No energy assumption can repair a transverse momentum contradiction.