Question

Your forehead can withstand a force of about 6.0 kn before fracturing, while your cheekbone can withstand only about 1.3 kn. suppose a 140 g baseball traveling at 30 m/s strikes your head and stops in 1.5 ms.
a. what is the magnitude of the force that stops the baseball?

b. what force does the baseball exert on your head? explain.

c. are you in danger of a fracture if the ball hits you in the fore- head? on the cheek?

Answer 1

a) The change in momentum of the ball is given by:

$\Delta p = m \Delta v$

where m=140 g=0.14 kg is the mass of the ball, and $\Delta v =30 m/s$ is the change in velocity. Substituting, we find

$\Delta p=(0.14 kg)(30 m/s)=4.2 kg m/s$

The ball stops in t=1.5 ms=0.0015 s; the magnitude of the force that stops the baseball is given by the ratio between the change in momentum and the time taken:

$F=\frac{\Delta p}{t}=\frac{4.2 kg m/s}{0.0015 s}=2800 N=2.8 kN$

b) The force we found at point a) is the force that the head exerts on the ball to stop it. However, Newton's third law states that when an object A exerts a force on an object B, object B also exerts a force equal and opposite on object A. If we apply this law to this case, we understand that the force exerted by the baseball on the head is equal to the force exerted by the head on the ball: therefore, the answer is still 2.8 kN.

c) The forehead is not in danger of a fracture, since it can withstand a maximum force of 6.0 kN, while the ball exerts a force of 2.8 kN. Instead, the cheek is in danger of fracture, because it can withstand only a maximum force of 1.3 kN.