In February 1955, a paratrooper fell 370 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 62 m/s (terminal speed), that his mass (including gear) was 71 kg, and that the magnitude of the force on him from the snow was at the survivable limit of 1.2 x 105 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?
(a): (minimum snow depth)*(maximum survivable force) = kinetic energy at impact = (1/2) M V^2
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(b) Impulse = momentum change = momentum at impact = M V
To answer this question, we need to use Newton's second law of motion, which states that the force applied to an object is equal to the rate of change of its momentum. Additionally, we can use the concept of impulse, which is the change in momentum of an object when a force acts upon it for a certain amount of time.
(a) To find the minimum depth of snow that would have stopped the paratrooper safely, we can use the concept of work-energy principle. According to the principle, the work done on an object is equal to its change in kinetic energy. Since the paratrooper's initial speed (before landing in the snow) is his terminal speed and his final speed is 0 m/s (as he comes to a stop), we can calculate the work done on him by the snow.
The work done by friction (snow) can be calculated using the equation:
Work = Force x Distance
Here, the force on the paratrooper from the snow is given as 1.2 x 10^5 N. The distance is the depth of snow. The work done on the paratrooper by the snow is equal to the change in kinetic energy of the paratrooper.
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
The initial kinetic energy can be calculated using the equation for kinetic energy:
Initial Kinetic Energy = (1/2) x mass x (initial velocity)^2
Substituting the given values:
Initial Kinetic Energy = (1/2) x 71 kg x (62 m/s)^2
The final kinetic energy is zero because the paratrooper comes to a stop.
Therefore,
Change in Kinetic Energy = 0 - [(1/2) x 71 kg x (62 m/s)^2]
Since the work done by the snow is equal to the change in kinetic energy, we can set it equal to the force on the paratrooper from the snow and solve for the depth of snow.
(1.2 x 10^5 N) x Distance = -(1/2) x 71 kg x (62 m/s)^2
Solving for Distance, we get:
Distance = (-(1/2) x 71 kg x (62 m/s)^2) / (1.2 x 10^5 N)
(b) The magnitude of impulse on the paratrooper from the snow can be calculated using the equation:
Impulse = Force x Time
Since the time of impact is not provided, we cannot directly calculate the impulse. However, we know that impulse is equal to the change in momentum. The change in momentum is given by:
Change in Momentum = Final Momentum - Initial Momentum
The initial momentum of the paratrooper can be calculated using the equation for momentum:
Initial Momentum = mass x (initial velocity)
Substituting the given values:
Initial Momentum = 71 kg x 62 m/s
The final momentum is zero because the paratrooper comes to a stop.
Therefore,
Change in Momentum = 0 - (71 kg x 62 m/s)
Thus, the magnitude of the impulse on the paratrooper from the snow is equal to the change in momentum:
Magnitude of Impulse = (71 kg x 62 m/s)
To solve this problem, we need to use the concept of impulse and momentum.
Step 1: Calculate the initial velocity of the paratrooper before falling.
We know that the final velocity (v) is 62 m/s since it is the terminal speed. The initial velocity (u) can be assumed to be 0 m/s since he started falling from rest.
Step 2: Calculate the time taken for the paratrooper to fall.
We can use the kinematic equation: v = u + at, where a is the acceleration due to gravity (approximately 9.8 m/s²). Solving for t, we get: t = (v - u) / a.
t = (62 - 0) / 9.8
t = 6.33 seconds.
Step 3: Calculate the impulse on the paratrooper.
Impulse (J) is defined as the change in momentum, given by J = m * Δv, where m is the mass and Δv is the change in velocity. Since the paratrooper's initial velocity was 0 m/s, Δv is equal to the terminal velocity of 62 m/s.
J = m * (v - u)
J = 71 kg * (62 m/s - 0 m/s)
J = 4422 Ns.
Step 4: Calculate the minimum depth of snow that would have stopped him safely.
The impulse experienced by the paratrooper is equal to the integral of the force over time. This can be represented as J = ∫ F(t) dt. Since the force exerted by the snow is constant, we can rewrite this as J = F * Δt, where Δt is the time taken for the snow to stop the paratrooper.
J = F * Δt
4422 Ns = (1.2 x 10^5 N) * Δt
Solving for Δt:
Δt = 4422 Ns / (1.2 x 10^5 N)
Δt ≈ 0.03685 seconds.
Step 5: Use the time taken to calculate the minimum depth of snow.
We can use the equation of motion, s = ut + (1/2)at^2, to calculate the minimum depth of snow (s), with u = 0 m/s, t = 0.03685 seconds, and a = 9.8 m/s².
s = (1/2) * a * t^2
s = (1/2) * 9.8 m/s² * (0.03685 s)^2
s ≈ 0.00675 meters, or 6.75 cm.
Therefore, the minimum depth of snow that would have stopped the paratrooper safely is approximately 6.75 cm, and the magnitude of the impulse on the paratrooper from the snow is approximately 4422 Ns.