A paratrooper fell 380m 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 58 m/s(terminal speed), that his mass (including gear) was 90 kg, and that the force on him from the snow was at the survvivable limist of 1.2 x 10^8 N
what is the minimum depth of snow that would have stopped him safetly
what is the magnitude of the impluse on him from the snow
Force * (Stopping distance)
= work done on the snow
= kinetic energy at impact
= (1/2) M V^2
X = (1/2) M V^2/(1.2 x 10^8 N)
The impulse, I, is equal to the momentum change
I = M V
To find the minimum depth of snow that would have stopped the paratrooper safely, we need to calculate the distance over which the paratrooper comes to a stop in the snow.
We can use the equation of motion: v² = u² + 2as, where:
- v = final velocity (0 m/s, as he stops)
- u = initial velocity (58 m/s, his terminal speed)
- a = acceleration (which we need to find)
- s = distance (which we want to find)
Rearranging the equation and substituting the given values:
0² = 58² + 2a(s)
0 = 3364 + 2as
We also know that the force acting on the paratrooper when he lands is given by F = ma, where:
- F = force (1.2 x 10^8 N, the survivable limit)
- m = mass (90 kg, including gear)
- a = acceleration (which we already used in the previous equation)
From the equation F = ma, we can isolate acceleration:
a = F / m
a = (1.2 x 10^8 N) / (90 kg)
a ≈ 1.33 x 10^6 m/s²
Now, we can substitute the value of acceleration (a) into the equation 0 = 3364 + 2as to solve for s:
0 = 3364 + 2(1.33 x 10^6 m/s²)(s)
0 = 3364 + 2.66 x 10^6 s
-3364 = 2.66 x 10^6 s
s ≈ -0.00127 m
Since the distance (s) cannot be negative, we discard the negative solution, so the minimum depth of snow required to stop the paratrooper safely is approximately 0.00127 meters (or 1.27 millimeters).
Now, let's calculate the magnitude of the impulse on the paratrooper from the snow.
Impulse (J) is defined as the change in momentum of an object and can be calculated using the equation J = Ft, where:
- J = impulse
- F = force (1.2 x 10^8 N, the survivable limit)
- t = time (which we need to find)
To find t, we'll use the equation of motion v = u + at, where:
- v = final velocity (0 m/s)
- u = initial velocity (58 m/s)
- a = acceleration (-1.33 x 10^6 m/s², negative because it opposes the motion)
- t = time
Rearranging the equation to solve for t:
0 = 58 + (-1.33 x 10^6 m/s²)t
1.33 x 10^6 m/s²t = 58
t = 58 / (1.33 x 10^6 m/s²)
t ≈ 4.36 x 10^-5 seconds
Now, we can substitute the values of force (F) and time (t) into the impulse equation:
J = (1.2 x 10^8 N)(4.36 x 10^-5 seconds)
J ≈ 5232 Newton-seconds, or 5232 Ns.
Therefore, the magnitude of the impulse on the paratrooper from the snow is approximately 5232 Ns.