91Ó°ÊÓ

Newton's Second Law is a fundamental concept in physics that relates the net force acting on an object to its mass and acceleration. It's expressed by the formula:

• \( F = ma \),

where \( F \) is the force applied to an object, \( m \) is the mass of the object, and \( a \) is the acceleration produced.
In our scenario involving the paratrooper, the snow applies a force to stop him from his fall. By rearranging the formula to solve for acceleration \( a = \frac{F}{m} \), we can determine how quickly the snow decelerates the paratrooper.
This acceleration (or rather, deceleration because it acts opposite to his motion) is key to understanding how the snow safely stops him.

Kinematic equations describe the mathematical relationships between displacement, velocity, acceleration, and time.

• These equations assume constant acceleration, which applies to this problem where deceleration is provided by snow.
• A key kinematic equation used is \( v^2 = u^2 + 2ad \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is acceleration, and \( d \) is displacement (or in this context, the depth of snow required to stop the paratrooper).

By setting \( v = 0 \) because he stops, and knowing \( u \) and \( a \), we can solve for the depth \( d \) of the snow.
This equation helps us connect how long the snow must decelerate the paratrooper to bring his velocity to zero.

Impulse is related to the change in momentum of an object. Momentum is the product of mass and velocity (\( p = mv \)). Impulse is defined as the change in momentum and is given by the equation:

• \( J = \Delta p = m\Delta v \),

where \( J \) is the impulse and \( \Delta v \) is the change in velocity.
In our problem, the paratrooper impacts the snow with an initial velocity of \( 56 \, \text{m/s} \) and comes to a stop, so \( \Delta v = 0 - 56 \, \text{m/s} \).
Impulse can also be expressed as the product of force and time of contact \( J = F \times \Delta t \), but in this case, we directly calculate it from momentum change.
Calculating impulse helps us understand the effect of the snow's force in bringing the paratrooper to rest.

Deceleration can be understood as negative acceleration that slows down an object. In the given scenario, the snow applies a force against the motion of the paratrooper, introducing deceleration.

• Using the formula \( a = \frac{F}{m} \), we find how much the velocity is reduced by force.
• This negative acceleration is crucial in determining how quickly the paratrooper can stop, which directly affects the depth of snow needed.

Deceleration links to the kinematic equations allowing us to calculate the required distance (or depth) for a safe stop.