Question

Determine the angle $\alpha$ formed by the string and the rod of mass $m_1$.

1.51. A ball moving at a velocity $v=10$ m/s hits the foot of a football player. Determine the velocity $u$ with which the foot should move for the ball impinging on it to come to a halt, assuming that the mass of the ball is much smaller than the mass of the foot and that the impact is perfectly elastic.

Answer 1

Part 1: Insufficient information to determine the angle. Part 2: The foot should move with a velocity of 5 m/s.

Answer 2

Part 1: Unfortunately, no additional details (free‐body diagram, forces, geometry, or additional masses) are provided regarding how the string is attached, what forces act, or under what equilibrium conditions the system is. Hence, the angle 𝛼 cannot be determined uniquely from the given information.

Part 2: Since the mass of the ball (m) is very small compared to that of the foot (M), we can assume that in the collision the foot’s velocity remains essentially unchanged. For an elastic collision where the ball comes to rest after the impact, the relative velocity of separation is equal to the relative velocity of approach. In the limit m ≪ M, it is known that

$v_{final(ball)} \approx -v_{ball} + 2u$

Setting $v_{final(ball)} = 0$ to have the ball come to rest gives

$0 = -v + 2u$
$2u = v$
$u = v/2$

Substituting $v = 10$ m/s:

$u = 10/2 = 5$ m/s

So, the foot must move with a velocity of 5 m/s.