A 72 kg man jumps fro a window 1.5 m above a sidewalk.
The acceleration of gravity is 9.81 m/s^2.
a) what is his speed just before his feet strike the pavement? Answer in units of m/s.
b) If the man jumps with his knees and ankles locked, the only cushion for his fall is approximately 0.49 cm in the pads of his feet.
Calculate the magnitude of the average force exerted on him by the ground in his situation. Answer in units of N.
To solve this problem, we can use the equations of motion and the principles of conservation of energy.
a) First, let's calculate the speed of the man just before his feet strike the pavement. We can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity (speed) of the man just before his feet strike the pavement (what we are trying to find)
u = initial velocity of the man (which is 0 since he is jumping vertically)
a = acceleration due to gravity (given as 9.81 m/s^2)
s = displacement (height of the jump, which is 1.5 m)
Plugging in the values, we have:
v^2 = 0^2 + 2 * 9.81 * 1.5
v^2 = 29.43
v = sqrt(29.43)
v ≈ 5.43 m/s
Therefore, the speed of the man just before his feet strike the pavement is approximately 5.43 m/s.
b) Next, let's calculate the magnitude of the average force exerted on the man by the ground. We can use the equation:
Work done = Force exerted * Distance
The work done by the average force exerted on the man is equal to the change in the potential energy of the man. The potential energy lost by the man equals the work done by the force exerted by the ground:
Potential energy lost = m * g * h
where:
m = mass of the man (given as 72 kg)
g = acceleration due to gravity (given as 9.81 m/s^2)
h = height of the cushion for the fall (given as 0.49 cm = 0.49/100 = 0.0049 m)
Plugging in the values, we have:
Potential energy lost = 72 * 9.81 * 0.0049
Potential energy lost ≈ 3.516 J
Since work done is equal to the potential energy lost, we have:
Work done = 3.516 J
Now, we know that work done is equal to force exerted times distance, and since the distance is the height of the cushion, we have:
Force exerted * 0.0049 = 3.516
Force exerted ≈ 3.516 / 0.0049
Force exerted ≈ 717.55 N
Therefore, the magnitude of the average force exerted on the man by the ground is approximately 717.55 N.
a) To determine the speed of the man just before his feet strike the pavement, we can use the principle of conservation of mechanical energy.
The potential energy of the man at the window can be calculated using the equation:
Potential Energy = mass × acceleration due to gravity × height
Potential Energy = 72 kg × 9.81 m/s^2 × 1.5 m
Next, we can convert the potential energy into kinetic energy just before his feet strike the pavement, assuming no energy is lost due to friction or air resistance. The equation for kinetic energy is:
Kinetic Energy = 0.5 × mass × velocity^2
Setting the potential energy equal to the kinetic energy, we have:
Potential Energy = Kinetic Energy
72 kg × 9.81 m/s^2 × 1.5 m = 0.5 × 72 kg × velocity^2
By rearranging the equation, we can solve for velocity:
velocity^2 = (2 × 9.81 m/s^2 × 1.5 m)
velocity = √(2 × 9.81 m/s^2 × 1.5 m)
Now, we can calculate the velocity:
velocity = √(29.43 m^2/s^2)
velocity ≈ 5.43 m/s
Therefore, the speed of the man just before his feet strike the pavement is approximately 5.43 m/s.
b) To calculate the magnitude of the average force exerted on the man by the ground, we can use Newton's second law of motion, which states that force is equal to mass times acceleration, or F = m × a.
First, we need to find the acceleration by using the equation:
acceleration = change in velocity / time
The change in velocity is the final velocity (0 m/s) minus the initial velocity (5.43 m/s). From the problem, we know the man's jump duration is not given, so let's assume it takes 0.1 seconds for him to reach the ground.
acceleration = (0 m/s - 5.43 m/s) / 0.1 s
acceleration ≈ -54.3 m/s^2
As the acceleration is negative, it indicates that the direction of the force exerted by the ground is upward.
Now we can substitute the values into the equation for force:
force = mass × acceleration
force = 72 kg × (-54.3 m/s^2)
force ≈ -3915.6 N
However, the magnitude of a force is always positive, so we can take the absolute value of the force to get the magnitude:
magnitude of force = |force| ≈ 3915.6 N
Therefore, the magnitude of the average force exerted on the man by the ground is approximately 3915.6 N.
v = g t
h = (1/2) g t^2
1.5 = 4.9 t^2 solve for t
then
v = 9.81 t
that is part a
.49 cm = .0049 meter
force * .0049 = (1/2) m v^2