A 85 kg pole-vaulter falls from rest from a height of 6.4 m onto a foam-rubber pad. The pole-vaulter comes to rest 0.48 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
a. V^2 = 0 + 19.8*6.4 = 126.72
V = 11.26 m/s.
To find the velocity of the pole-vaulter just before reaching the pad, we can use the kinematic equation:
v = u + at
where:
v = final velocity (unknown)
u = initial velocity (0 m/s, because the pole-vaulter falls from rest)
a = acceleration (unknown)
t = time taken to come to rest (0.48 s)
Since the pole-vaulter comes to rest, the final velocity (v) will be 0 m/s.
0 = 0 + a * 0.48
Simplifying the equation, we find:
0 = 0.48a
Therefore,
a = 0 m/s²
Now, we can use another kinematic equation:
v = u + at
0 = 0 + 0 * t
So, the velocity of the pole-vaulter just before reaching the pad is 0 m/s downward.
To calculate the constant force exerted on the pole-vaulter due to the collision, we can use Newton's second law:
F = m * a
where:
F = force (unknown)
m = mass of the pole-vaulter (85 kg)
a = acceleration (0 m/s², because the pole-vaulter comes to rest)
Substituting the values into the equation, we get:
F = 85 kg * 0 m/s²
Therefore, the constant force exerted on the pole-vaulter due to the collision is 0 N upward.
To solve this problem, we can use the principles of kinematics and Newton's laws of motion. Let's break it down step by step:
Step 1: Calculate the athlete's initial potential energy:
The potential energy of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
PE = (85 kg)(9.8 m/s^2)(6.4 m) = 5331.2 J
Step 2: Calculate the athlete's final kinetic energy:
The kinetic energy of an object is given by the formula KE = 1/2mv^2, where v is the velocity. Since the athlete comes to rest, the final kinetic energy is zero.
KE = 0
Step 3: Calculate the work done by the foam-rubber pad:
The work done by a force is given by the formula W = Fd, where F is the force and d is the distance. In this case, the force exerted by the pad is in the opposite direction to the displacement of the athlete, so the work is negative. The distance can be calculated as the height from which the athlete falls (6.4 m). Therefore, the work done is:
W = -PE = -5331.2 J
Step 4: Calculate the average force exerted on the pole-vaulter:
The average force can be found using the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. Since the final kinetic energy is zero, the work done by the foam-rubber pad is equal to the initial kinetic energy:
-5331.2 J = 1/2(85 kg)v^2
v^2 = (-5331.2 J * 2) / (85 kg)
v^2 = -125.44 m^2/s^2 (note that this is negative)
Step 5: Calculate the velocity just before reaching the pad:
The velocity is the square root of the squared value found in the previous step. Since the velocity is a scalar quantity, we take the positive square root:
v = sqrt(-125.44 m^2/s^2) = 11.2 m/s downward (note that the answer is negative because the velocity is directed downward)
Therefore, the athlete's velocity just before reaching the pad is 11.2 m/s downward (a).
Step 6: Calculate the constant force exerted on the pole-vaulter due to the collision:
Since the athlete comes to rest, the change in velocity is v = 11.2 m/s downward. The time taken to come to rest is given as 0.48 s.
Using Newton's second law, which states that force (F) is equal to mass (m) multiplied by acceleration (a), we can calculate the acceleration:
v = at
a = v / t
a = (-11.2 m/s) / (0.48 s)
a = -23.33 m/s^2 (note that this is negative)
The force exerted on the pole-vaulter due to the collision is then given by:
F = ma
F = (85 kg)(-23.33 m/s^2)
F = -1983 N (note that the answer is negative because the force is directed upward)
Therefore, the constant force exerted on the pole-vaulter due to the collision is 1983 N upward (b).