91Ó°ÊÓ

Newton's Second Law of Motion states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. This can be written as the formula:
\( F = ma \).

In our problem, we know the force acting on the paratrooper from the snow is at the survivable limit of \( 1.2 \times 10^5 \) N, and the paratrooper's mass (including gear) is 85 kg. Using this information, we can determine the deceleration the paratrooper experienced upon hitting the snow.

By rearranging Newton's Second Law to solve for acceleration (\( a \)), we get:
\( a = \frac{F}{m} \).

Substituting the known values, we find the deceleration to be \( 1411.76 \text{ m/s}^2 \). This deceleration can tell us how quickly the paratrooper slowed down after hitting the snow. Newton's Second Law is crucial in analyzing the forces and motion involved in this scenario.

Kinematic equations help us describe the motion of objects by relating variables such as displacement (\( d \)), initial velocity (\( u \)), final velocity (\( v \)), acceleration (\( a \)), and time (\( t \)).

For our problem, we use one of the kinematic equations to find the necessary depth of the snow to safely stop the paratrooper. The relevant equation is:
\( v^2 = u^2 + 2ad \).

Here, the final velocity (\( v \)) is 0 since the paratrooper comes to a stop. The initial velocity (\( u \)) is 56 m/s, and the acceleration (\( a \)) is the deceleration we calculated earlier (\( -1411.76 \text{ m/s}^2 \)). Plugging in these values, we solve for the displacement (\( d \)), which represents the minimum depth of the snow:

\(0 = 56^2 + 2(-1411.76)d \)
\(0 = 3136 - 2823.52d \)
\(2823.52d = 3136 \)
\( d = \frac{3136}{2823.52} \approx 1.11 \text{ m} \).

Therefore, the snow must be at least 1.11 meters deep to stop the paratrooper safely.

Impulse and momentum are key concepts in understanding how forces affect motion over time. Momentum (\( p \)) is defined as the product of an object's mass and its velocity, expressed by \( p = mv \).

Impulse (\( J \)) is the change in momentum, which can be caused by a force acting over a period of time. The relationship is given by:
\( J = \text{Change in momentum} \).

In our problem, we calculate the impulse on the paratrooper from the snow by considering his change in velocity upon impact. The formula used is:
\( J = \text{mass} \times \text{change in velocity} \) or \( J = m \times (v - u) \)

Given his mass (85 kg) and considering his initial velocity (56 m/s) and his final velocity (0 m/s), we find:

\( J = 85 \times (0 - 56) = -4760 \text{ kg} \times \text{m/s} \).

The negative sign indicates that the impulse is in the direction opposite to his initial velocity. The magnitude of the impulse is \( |J| = 4760 \text{ kg} \times \text{m/s} \).

Understanding impulse and momentum helps explain how the paratrooper's high-speed impact with the snow was suddenly reduced to zero speed, safely bringing him to rest.