In February 1955, a paratrooper fell 365 m from an airplane without being able to open his chute but happened to land in snow, suffering only minor injuries. Assume that his speed at impact was 60 m/s (terminal speed), that his mass (including gear) was 90 kg, and that the force on him from the snow was at the survivable limit of 120000 N. What are (a) the minimum depth of snow that would have stopped him safely and (b) the magnitude of the impulse on him from the snow?
To find the minimum depth of snow that would have stopped the paratrooper safely and the magnitude of the impulse on him from the snow, we can use the principles of physics such as the equation of motion and impulse-momentum theory.
(a) Minimum depth of snow:
The minimum depth of snow required to stop the paratrooper safely can be calculated using the equation for distance traveled during deceleration:
d = (v^2 - u^2) / (2a)
where
d = distance traveled
v = final velocity (0 m/s, as the paratrooper stopped)
u = initial velocity (60 m/s, the terminal speed)
a = acceleration
In this case, the acceleration is caused by the force exerted on the paratrooper by the snow. The force can be calculated using Newton's second law:
F = ma
where
F = force (120,000 N)
m = mass of the paratrooper (90 kg)
a = acceleration
From these equations, we can rearrange to solve for the minimum depth of snow required:
d = (0^2 - 60^2) / (2 * a)
120,000 = 90 * a
Solving for a, we get:
a = 120,000 / 90
a = 1,333.33 m/s^2
Substituting this value of acceleration into the equation for distance traveled, we can find the minimum depth of snow:
d = (-60^2) / (2 * 1,333.33)
d = -3600 / 2666.67
d ≈ -1.35 m
Since distance cannot be negative, the minimum depth of snow required to stop the paratrooper safely is approximately 1.35 meters.
(b) Magnitude of impulse:
Impulse is defined as the change in momentum of an object and is equal to the force applied multiplied by the time over which it acts. In this case, the impulse on the paratrooper is equal to the force exerted by the snow multiplied by the time it takes for the paratrooper to come to a stop.
Impulse = Force * Time
To find the time, we need to determine the distance traveled during deceleration and the average velocity before coming to a stop.
Using the equation:
v = u + at
where
v = final velocity (0 m/s)
u = initial velocity (60 m/s)
a = acceleration
We can rearrange the equation to solve for time:
t = (v - u) / a
t = (0 - 60) / (-1333.33)
t ≈ 0.045 s
Now, we can calculate the impulse:
Impulse = Force * Time
Impulse = 120,000 * 0.045
Impulse ≈ 5,400 Ns
Therefore, the magnitude of the impulse on the paratrooper from the snow is approximately 5,400 Newton-seconds (Ns).
To find the minimum depth of snow that would have stopped the paratrooper safely, we can use the concept of impulse. Impulse is defined as the product of force and time and is equal to the change in momentum.
(a) The minimum depth of snow can be calculated using the equation:
Impulse = Force * Time
Impulse = change in momentum
Since the paratrooper comes to a stop in the snow, his initial momentum is equal to zero.
Initial momentum = mass * initial velocity
Final momentum = mass * final velocity
The change in momentum is equal to the final momentum.
change in momentum = final momentum - initial momentum
Given:
Mass (including gear) = 90 kg
Initial velocity = 60 m/s (terminal speed)
Force = 120000 N
To find the time of impact, we can use the equation: Final velocity = initial velocity + acceleration * time
Final velocity = 0 m/s (The paratrooper comes to a stop)
Initial velocity = 60 m/s
Acceleration = (Force / mass)
Rearranging the equation, we can solve for time:
Final velocity = initial velocity + (Force / mass) * time
0 = 60 + (120000 / 90) * time
Solving for time:
-60 = (120000 / 90) * time
time = -60 * (90 / 120000)
time = -0.045 seconds (ignoring the negative sign)
Now we can calculate the impulse:
Impulse = Force * Time
Impulse = 120000 * 0.045
Impulse = 5400 N*s
Since impulse is equal to the change in momentum, we can calculate the final momentum:
Impulse = change in momentum
change in momentum = final momentum - initial momentum
Given:
Mass (including gear) = 90 kg
Initial velocity = 60 m/s (terminal speed)
Initial momentum = mass * initial velocity
Initial momentum = 90 kg * 60 m/s
change in momentum = 5400 N*s
Final momentum = change in momentum + initial momentum
Final momentum = 5400 N*s + (90 kg * 60 m/s)
Now we can calculate the minimum depth of snow that would have stopped the paratrooper safely:
The stopping distance can be calculated using the equation:
Stopping distance = (final momentum^2 - initial momentum^2) / (2 * net force)
Given:
Force = 120000 N
Stopping distance = (final momentum^2 - initial momentum^2) / (2 * net force)
substituting the values:
Stopping distance = (Final momentum^2 - Initial momentum^2) / (2 * Force)
Stopping distance = ((5400 N*s + (90 kg * 60 m/s))^2 - (90 kg * 60 m/s)^2) / (2 * 120000 N)
Calculating the values:
Stopping distance = (5400^2 + (90 * 60)^2 - (90 * 60)^2) / (2 * 120000)
Stopping distance = (5400^2) / (2 * 120000)
Stopping distance = 29160000 / 240000
Stopping distance = 121 m
Therefore, the minimum depth of snow that would have stopped the paratrooper safely is 121 meters.
(b) The magnitude of the impulse on the paratrooper from the snow is 5400 N*s.
(b) The impulse the snow had to exert on him equaled the momentuum of him and his gear at impact:
M*V = 90 kg*60 m/s = 5400 kg m/s
(a) For the mimimum snow depth needed to avoid a lethal force greater than 120,000N
Kinetic energy @ impact = 120,000*depth
= (1/2)*90*65^2 = 190,000 J
Depth = 190,000/120,000 = 1.6 meters