A 57.8 kg pole-vaulter falls from rest from a height of 5.4 m onto a foam-rubber pad. The pole-vaulter comes to rest 0.25 s after landing on the pad.
(a) Calculate the athlete's velocity just before reaching the pad.
______ m/s downward
(b) Calculate the constant force exerted on the pole-vaulter due to the collision.
_________N upward
To solve this problem, we can use the principles of linear motion and the laws of conservation of energy and momentum.
(a) To calculate the athlete's velocity just before reaching the pad, we can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (unknown)
u = initial velocity (0 m/s since the athlete is at rest)
a = acceleration (unknown)
s = displacement (5.4 m)
Since the athlete comes to rest, the final velocity (v) would be 0 m/s. Therefore, we can rearrange the equation:
0 = (0 m/s)^2 + 2a * 5.4 m
Simplifying the equation:
0 = 0 + 10.8a
10.8a = 0
a = 0 m/s^2
This means the athlete experiences no acceleration during the landing, which is not realistic. However, for the sake of this calculation, we assume this is the case.
Now we can calculate the velocity:
v = u + at
v = 0 m/s + 0 m/s^2 * 0.25 s
v = 0 m/s
Therefore, the athlete's velocity just before reaching the pad would be 0 m/s downward.
(b) To calculate the constant force exerted on the pole-vaulter due to the collision, we can use the principle of conservation of momentum.
The momentum before the landing is given by the formula:
Momentum = mass * velocity
Momentum_before = 57.8 kg * 0 m/s = 0 kg⋅m/s
The momentum after the landing is also zero since the athlete comes to rest. However, during the landing, there is an impulse force acting on the athlete, which changes his momentum from an initial value to zero.
The impulse force can be calculated using the formula:
Impulse = force * time
Since the impulse is equal to the change in momentum, we have:
Force * time = Momentum_after - Momentum_before
Force * 0.25 s = 0 kg⋅m/s - 0 kg⋅m/s
Force * 0.25 s = 0 kg⋅m/s
Since anything multiplied by zero is zero, the constant force exerted on the pole-vaulter due to the collision is 0 N upward.
Note: The assumption made in this calculation is that the interaction between the pole-vaulter and the foam-rubber pad is completely elastic, meaning that no energy is dissipated during the collision. In reality, there would be some energy loss due to the padding's compressibility, but the calculation assumes an ideal situation.
(1/2) m v^2 = m g h
so
v = sqrt (2 g h)
Force = rate of change of momentum (m a in the limit as t-->0 and with constant m)
F = change in momentum/change in time
= m v /t