In one day, a $75$-kg mountain climber ascends from the $1500$-m level on a vertical cliff to the top at $2400$ m. The next day, she descends from the top to the base of the cliff, which is at an elevation of $1350$ m. What is her change in gravitational potential energy on the first day?

Verified step by step guidance

1. Step 1: Recall the formula for gravitational potential energy, which is given by: $U=mgh$, where $U$ is the gravitational potential energy, $m$ is the mass of the object, $g$ is the acceleration due to gravity (approximately 9.8 m/s²), and $h$ is the height above a reference point.
2. Step 2: Calculate the change in gravitational potential energy for the first day. The change in gravitational potential energy is given by: $\mathrm{\Delta U}=mg(h$_f - h_i), where $h$_f is the final height and $h$_i is the initial height.
3. Step 3: Substitute the given values into the formula. The mass of the climber is $75$ kg, the acceleration due to gravity is $9.8$ m/s², the initial height is $1500$ m, and the final height is $2400$ m.
4. Step 4: Perform the subtraction for the height difference: $h$_f - h_i = 2400 - 1500 m. This gives the height difference.
5. Step 5: Multiply the mass, acceleration due to gravity, and the height difference to find the change in gravitational potential energy: $\mathrm{\Delta U}=75\times 9.8\times \; (2400-1500)$. This will give the change in gravitational potential energy for the first day.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula GPE = mgh, where m is mass, g is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and h is the height above a reference point. In this scenario, the climber's change in height directly affects her GPE.

Change in Height

The change in height is a critical factor in determining the change in gravitational potential energy. It is calculated by subtracting the initial height from the final height. In the given problem, the climber ascends from 1500 m to 2400 m, resulting in a height change of 900 m, which is essential for calculating the GPE change.

Mass and Weight

Mass is a measure of the amount of matter in an object, typically measured in kilograms. Weight, on the other hand, is the force exerted by gravity on that mass, calculated as weight = mass × gravity. In this problem, the climber's mass of 75 kg is used to determine the gravitational potential energy change as she ascends the cliff.