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 88.1-kg man just before contact with the ground has a speed of 3.53 m/s. (a) In a stiff-legged landing he comes to a halt in 3.56 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 0.199 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).
force = rate of change of momentum
= 88.1*3.53 / .00356 = 87,358 N
= 88.1*3.53/.199 = 1,563 N
with gravity add 88.1*9.81 = 864 N
Hey, wait a minute.
If there were no gravity it would have taken less time to stop.
The total force up including gravity = change of momentum so there is no change to the force in part b due to gravity.
To solve this problem, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration. In this case, we can assume that the acceleration is constant during the landing.
(a) To find the magnitude of the average net force in a stiff-legged landing, we first need to find the acceleration. We can use the equation of motion:
v = u + at,
where v is the final velocity (0 m/s in this case), u is the initial velocity (3.53 m/s), a is the acceleration, and t is the time taken to come to a halt (3.56 ms = 0.00356 s).
Substituting the known values into the equation, we have:
0 = 3.53 + a(0.00356).
Solving for the acceleration a, we get:
a = -3.53 / 0.00356.
Now, we can calculate the magnitude of the average net force using Newton's second law:
F = m * a,
where F is the net force, m is the mass of the man (88.1 kg), and a is the acceleration.
Substituting the known values into the equation, we get:
F = 88.1 * (-3.53 / 0.00356).
(b) To find the magnitude of the average net force when the man bends his knees, we can use the same approach as in part (a), but with a different time taken to come to a halt (0.199 s).
Using the equation of motion:
0 = 3.53 + a(0.199),
we can solve for the acceleration a:
a = -3.53 / 0.199.
Then, calculate the magnitude of the average net force using Newton's second law:
F = 88.1 * (-3.53 / 0.199).
(c) In this case, we need to consider the direction of the forces. The force due to gravity points downward and has a magnitude of:
F_gravity = m * g,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
The force applied by the ground on the man, which opposes the force due to gravity, has an equal magnitude but opposite direction.
Therefore, the magnitude of the force applied by the ground in part (b) is:
|F_applied| = |F_gravity| = m * g.
Substituting the known values into the equation, we get:
|F_applied| = 88.1 * 9.8.
Note that the magnitude of the force applied by the ground on the man will be larger in the stiff-legged landing (part a) than in the knee-bending landing (part b) since the time taken to come to a halt is shorter in part a.