A 220 lb running back is running along the sideline of the football field toward the goal line. The running back is 1.0 m from the sideline running at 8.0 m/s toward the goal. A 180 lb defender runs across the field toward the running back 2.0 m from the goal line. Just before the collision, both players leap into the air. The running back keeps his horizontal speed of 8.0 m/s. What is the minimum possible horizontal velocity for the defender in order to ensure the players cross out of bounds before crossing the goal line?
System: The running back and the defender, treated as point particles. External influences are negligible compared to the internal forces of the collision, and so are neglected.
Model: [Momentum and Impulse].
Approach: We are interested only in the horizontal components of the velocity. If we treat the players as point masses, then the defender's goal is for the horizontal velocity after the collision to be directed at 30 degrees from the running back's original direction of motion, as shown in the figure.
PICTURE
Since we are assuming external forces are negligible during the collision, we will have constant horizontal momentum:
\begin
[ p_
^
=p_
^
+p_
^
= p_
^
= p_
^
+p_
^
] [p_
^
=p_
^
+p_
^
= p_
^
= p_
^
+p_
^
] \end
where RB denotes the running back and DF the defender. For the case at hand, these equations can be simplified to read:
\begin
[ m^
v_
^
= (m^
+m^
)v_
]
[ m^
v_
^
= (m^
+m^
) v_
] \end
We can immediately solve the x direction equation:
\begin
[ v_
= \frac{m^
v_
{RB}}{m
+m^{DF}} = ]\end
but the y equation contains two unknowns. We can resolve this problem by using the constraints (discussed above) that in order to end up out of bounds:
\begin
[ v_
= v_
\tan 30^
= ]\end
With this information, we can solve to find:
\begin
[ v_
^
= \frac{m^
+m^{DF}}{m^{DF}}v_
= ...] \end
so that the horizontal component of the defender's velocity must be at least ....