91Ó°ÊÓ

When discussing the motion of the skier, kinetic energy plays a pivotal role. Kinetic energy (\( KE \)) is the energy that an object possesses due to its motion. It is determined by the formula:

• \( KE = \frac{1}{2} mv^2 \)

where \( m \) is the mass of the skier, and \( v \) is the velocity at which the skier is moving.
In our exercise, the skier starts with an initial speed of 24 m/s, leading to an initial kinetic energy calculation as follows:

• \( KE_i = \frac{1}{2} \times 60 \mathrm{~kg} \times (24 \mathrm{~m/s})^2 = 17280 \mathrm{~J} \)

Upon landing, the skier's speed reduces to 22 m/s, changing the kinetic energy to:

• \( KE_f = \frac{1}{2} \times 60 \mathrm{~kg} \times (22 \mathrm{~m/s})^2 = 14520 \mathrm{~J} \)

This reduction in kinetic energy highlights how velocity plays a significant role in determining the energy of moving objects like our skier.

Potential energy (\( PE \)) refers to the energy held by an object due to its position relative to other objects. In the context of our skier, it is primarily gravitational potential energy. The formula to determine potential energy is:

• \( \Delta PE = mgh \)

where \( m \) is the mass of the skier, \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height difference the skier experiences.
For the skier, the change in height upon landing is 14 m below the end of the ramp, which results in a potential energy change calculated as:

• \( \Delta PE = 60 \mathrm{~kg} \times 9.8 \mathrm{~m/s}^2 \times 14 \mathrm{~m} = 8232 \mathrm{~J} \)

In this scenario, the decrease in potential energy indicates the work done by gravity as the skier descends.

Air drag, or air resistance, acts against the motion of the skier, causing energy loss. This effect is part of the reason why the skier's velocity decreases as they descend, alongside gravitational forces. By calculating the total mechanical energy change, we can quantify the energy lost owing to air drag.
The initial total mechanical energy (\( E_i \)) is equivalent to the sum of the initial kinetic energy and initial potential energy. In this example:

• \( E_i = KE_i + PE_i = 17280 \mathrm{~J} \)

The final mechanical energy (\( E_f \)) is the combination of the final kinetic energy and the negative potential energy change.

• \( E_f = 14520 \mathrm{~J} - 8232 \mathrm{~J} = 6288 \mathrm{~J} \)

Therefore, the energy reduced due to air drag can be calculated by:

• \( \Delta E = E_i - E_f = 17280 \mathrm{~J} - 6288 \mathrm{~J} = 10992 \mathrm{~J} \)

This reduction highlights how air resistance plays a crucial role in energy dynamics by influencing the skier's final speed and overall energy during the jump.