Final answer:
The force exerted by a boxing glove on an opponent's face and the force exerted by a bare knuckle are calculated using the work-energy principle and considering the distances over which the face and the glove or knuckles compress. The force with the boxing glove is found to be lower due to the larger compression distance, although it may still cause injury.
Step-by-step explanation:
To calculate the force exerted by a boxing glove on an opponent's face, we need to use the work-energy principle which states that the work done is equal to the change in kinetic energy. Assuming the boxer's arm and glove are a single object coming to a stop, the initial kinetic energy (KE) can be found using the equation KE = ½ mv², where m is the mass of the arm and glove, and v is the initial speed.
(a) The initial kinetic energy of the 7.00-kg arm and glove moving at 10.0 m/s is KE = ½ (7.00 kg)(10.0 m/s)². To bring this to rest, the work done would be equal to this KE and the work done (W) is equal to the force (F) times the distance (d): W = Fd. The glove and face compress 7.50 cm, so d = 0.075 m. So, F = KE / d. Plugging in the numbers, we calculate the force exerted by the glove.
(b) Similarly, for a blow with no gloves (bare knuckles), the compression distance is only 2.00 cm or 0.020 m. Using the same method, we can calculate the force exerted in this scenario. The force will be higher in this case because the compression distance is shorter, assuming the same amount of initial kinetic energy.
(c) The magnitude of the force with the glove on is lower than without the glove due to the greater compression distance, which allows for a longer time to decelerate the arm and glove, resulting in a smaller force over the impact. However, the force with the glove can still be high enough to cause damage, suggesting that while gloves do reduce the impact, they do not completely eliminate the risk of injury.