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 69.3-kg man just before contact with the ground has a speed of 6.94 m/s. (a) In a stiff-legged landing he comes to a halt in 1.01 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 0.267 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)..
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To solve this problem, we can use the equations of motion to find the average net force acting on the man during both landings.
(a) In a stiff-legged landing, the man comes to a halt in 1.01 ms. We can use the equation of motion:
v = u + at
where:
- v is the final velocity (0 m/s, since the man comes to a halt)
- u is the initial velocity (6.94 m/s)
- a is the acceleration
- t is the time (1.01 ms or 0.00101 s)
Rearranging the equation, we have:
a = (v - u) / t
a = (0 - 6.94) / 0.00101
a = -6861.39 m/s^2
Now, we can use Newton's second law of motion:
F = ma
F = (69.3 kg) * (-6861.39 m/s^2)
F ≈ -475,909.267 N
The magnitude of the average net force in a stiff-legged landing is approximately 475,909.267 N.
(b) When the man bends his knees, he comes to a halt in 0.267 s. Following the same steps as above, we find:
a = (v - u) / t
a = (0 - 6.94) / 0.267
a = -26.006 m/s^2
F = ma
F = (69.3 kg) * (-26.006 m/s^2)
F ≈ -1,801.397 N
The magnitude of the average net force when the man bends his knees is approximately 1,801.397 N.
(c) In a stiff-legged landing, only the force due to gravity acts on the man (downward). Therefore, the magnitude of the force applied by the ground on the man is equal to the magnitude of the force due to gravity.
The force due to gravity is given by:
Fg = mg
Fg = (69.3 kg) * (9.8 m/s^2) [acceleration due to gravity]
Fg ≈ 678.54 N
Therefore, the magnitude of the force applied by the ground on the man when he bends his knees is approximately 678.54 N.
To solve this problem, we will use Newton's second law, which 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 1.01 ms. We can use the equation of motion: final velocity (vf) = initial velocity (vi) + acceleration (a) * time (t)
Since the man comes to a halt, his final velocity (vf) is 0 m/s, and his initial velocity (vi) is 6.94 m/s. Therefore, the acceleration (a) can be calculated as follows:
vf = vi + a * t
0 = 6.94 + a * 1.01 ms
To convert the time to seconds, we divide by 1000:
0 = 6.94 + a * 0.00101 s
Now we can solve for the acceleration (a):
a = (0 - 6.94) / 0.00101
a ≈ -6861.4 m/s²
The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, which is consistent with coming to a halt.
Now we can calculate the magnitude of the average net force using Newton's second law:
F = m * a
Given that the mass (m) of the man is 69.3 kg:
F = 69.3 kg * (-6861.4 m/s²)
F ≈ -474497 N
Therefore, the magnitude of the average net force that acts on the man during this time is approximately 474497 N.
(b) When the man bends his knees, he comes to a halt in 0.267 s. Using the same process as in part (a), we can calculate the acceleration (a):
vf = vi + a * t
0 = 6.94 + a * 0.267 s
Solving for the acceleration (a):
a = (0 - 6.94) / 0.267
a ≈ -26 m/s²
Now we can calculate the magnitude of the average net force:
F = m * a
F = 69.3 kg * (-26 m/s²)
F ≈ -1799.8 N
Therefore, the magnitude of the average net force when the man bends his knees is approximately 1799.8 N.
(c) In this part, we need to consider both the upward force applied by the ground and the downward force due to gravity.
The man comes to a halt, so the net force is zero. This means that the magnitude of the upward force applied by the ground must equal the magnitude of the downward force due to gravity.
Considering the downward force:
Weight = m * g
Where m is the mass of the man (69.3 kg), and g is the acceleration due to gravity (approximately 9.8 m/s²).
Weight = 69.3 kg * 9.8 m/s²
Weight ≈ 679.14 N
Since the net force is zero, the magnitude of the upward force applied by the ground is also 679.14 N.
Therefore, the magnitude of the force applied by the ground on the man when he bends his knees (part b) is approximately 679.14 N.