When jumping straight down, you can be seriously injured if you land stiff-legged. One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. A 68.5-kg man just before contact with the ground has a speed of 5.59 m/s. (a) In a stiff-legged landing he comes to a halt in 1.64 ms. Find the magnitude of the average net force that acts on him during this time. (b) When he bends his knees, he comes to a halt in 0.290 s. Find the magnitude of the average net force now. (c) During the landing, the force of the ground on the man points upward, while the force due to gravity points downward. The average net force acting on the man includes both of these forces. Taking into account the directions of the forces, find the magnitude of the force applied by the ground on the man in part (b).
To find the answers to each of the sub-questions, we can make use of Newton's second law of motion. This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
(a) In a stiff-legged landing, the man comes to a halt in 1.64 ms. We can find the acceleration by using the formula: acceleration = change in velocity / time taken.
Given:
Mass of the man, m = 68.5 kg
Initial velocity, u = 5.59 m/s
Final velocity, v = 0 m/s
Time taken, t = 1.64 ms = 1.64 × 10^-3 s
Using the formula for acceleration: acceleration = (v - u) / t
Substituting the values: acceleration = (0 - 5.59) / (1.64 × 10^-3)
Calculating the acceleration, we find: acceleration = -3400 m/s^2 (negative sign indicates deceleration)
Now, we can find the magnitude of the average net force by multiplying the mass of the man with the acceleration:
Average net force = mass × acceleration = 68.5 kg × (-3400 m/s^2)
Calculating the magnitude of the force, we get: Average net force = 232,900 N
(b) In a kneed-bend landing, the man comes to a halt in 0.290 s. We can follow the same steps as in part (a) to find the acceleration and then the magnitude of the average net force.
Given:
Time taken, t = 0.290 s
Using the formula for acceleration: acceleration = (v - u) / t
Substituting the values: acceleration = (0 - 5.59) / 0.290
Calculating the acceleration, we find: acceleration = -19.2078 m/s^2 (again, negative sign indicates deceleration)
Now, we can find the magnitude of the average net force by multiplying the mass of the man with the acceleration:
Average net force = mass × acceleration = 68.5 kg × (-19.2078 m/s^2)
Calculating the magnitude of the force, we get: Average net force = 1,316.33 N
(c) In part (b), we found the magnitude of the average net force when the man bends his knees. This force is the sum of the force applied by the ground on the man upward and the force due to gravity acting downward.
Since the man comes to a halt in his motion, the magnitude of the average net force is equal to the magnitude of the force applied by the ground on the man.
Using the value calculated in part (b), the magnitude of the force applied by the ground on the man = 1,316.33 N.