Question

When a ball of mass $m$ is dropped from rest from a height $h$, its kinetic energy just before landing is KE. Now, suppose a second ball of mass $4m$ is dropped from rest from a height $h/4$. (a) Just before ball 2 lands, is its kinetic energy $4KE,2KE,KE,KE/2$, or $KE/4$ ? (b) Choose the best explanation from among the following: A. The two balls have the same initial energy. B. The more massive ball will have the greater kinetic energy. C. The lower drop height results in a reduced kinetic energy.

Short answer

(a) KE; (b) C.

Step-by-step solution

Compute the Kinetic Energy for Ball 1

For the first ball, the kinetic energy just before it lands is given by the gravitational potential energy at height $h$. Thus, the kinetic energy $KE=mgh$, where $m$ is the mass and $g$ is the acceleration due to gravity.

Analyze Parameters for Ball 2

The second ball has a mass $4m$ and is dropped from a height $\frac{h}{4}$. We will calculate its kinetic energy just before landing to compare it with Ball 1.

Compute Kinetic Energy for Ball 2

For the second ball, substitute its mass and height into the potential energy formula: $K{E}_2=4m\cdot g\cdot \frac{h}{4}=mgh$. Thus, the kinetic energy of Ball 2 is $KE$, which is the same as Ball 1's kinetic energy.

Decide on the Correct Kinetic Energy Option

Based on the calculations, size up the options: $K{E}_2=KE$, so the correct answer is $KE$.

Evaluate Explanation Choices

Analyze each explanation: - A: Incorrect, the initial energies are different due to different masses and heights. - B: Incorrect, the kinetic energy depends on both mass and height, not just mass. - C: Correct, the reduced height influences the energy resulting in kinetic energy equivalent to Ball 1.

Key concepts

Gravitational Potential Energy

Gravitational potential energy is the energy that an object possesses due to its position in a gravitational field. It depends on three key things: the mass of the object ($m$), the height ($h$) from which the object is dropped, and the acceleration due to gravity ($g$), which is approximately 9.81m/s² on Earth.

The formula to calculate the gravitational potential energy ($PE$) is:

• $PE=mgh$

This formula indicates that a heavier object or an object placed higher up has more potential energy. When the ball starts at rest and is dropped, its potential energy turns into kinetic energy.

In the exercise, both balls start with potential energy and convert it all to kinetic energy as they fall. Despite different starting parameters, their potential energy transitions according to their specific mass and height details.

Mass and Height Relationship

The relationship between mass and height is crucial in understanding how kinetic energy is influenced when an object falls. Mass ($m$) and height ($h$) are tightly connected in the energy equation.

In the given problem, Ball 1 has a mass of $m$, higher than Ball 2, which has mass $4m$. However, Ball 2 is dropped from a height of $\frac{h}{4}$.

• For Ball 1, all potential energy ($mgh$) converts to kinetic energy.
• Ball 2's potential energy is calculated as $4m\cdot g\cdot \frac{h}{4}$, simplifying back to $mgh$, showing equal balances despite differing values.

This comparison underlines how mass and height work in tandem to shape potential and kinetic energies. Larger mass can balance out lower height, resulting in similar energy transfers.

Energy Conservation

Energy conservation is a fundamental concept in physics where the total energy in an isolated system remains constant. For dropping objects, gravitational potential energy transforms into kinetic energy during the fall.

As both balls in this exercise are dropped, their potential energy converts completely into kinetic energy as they reach the ground. This conversion doesn't add or lose energy—it's simply transformed from one type to another.

• For Ball 1, the same gravitational potential energy ($mgh$) at height $h$ turns into kinetic energy.
• Similarly, Ball 2's potential energy ($4m\cdot g\cdot \frac{h}{4}$) leads to an equal kinetic energy value when simplified.

The conservation principle ensures that the amount of energy before the drop is equivalent to the energy just before landing, thus explaining why Ball 2's kinetic energy matches that of Ball 1.