Question

An object falls from rest from a height $h$. Determine the kinetic energy and the potential energy of the object as a function $(a)$ of time and $(b)$ of height. Graph the expressions and show that their sum - the total mechanical energy - is constant in each case.

Short answer

Potential energy as a function of time is given by $PE(t)=mg(h-0.5g{t}^2)$, kinetic energy over time is $KE(t)=0.5m(gt{)}^2$, potential energy as a function of height is $PE(h)=mg(h-s)$, kinetic energy over height is $KE(h)=mgs$. The constant mechanical energy validates the law of conservation of energy.

Step-by-step solution

Write down the known variables

From the problem, initial velocity $u=0$ (object at rest), gravitational acceleration $g=9.8m/s^2$ (considering the direction of fall as positive), initial height $h$, velocity at time $t$ as $v$ and height fallen within time $t$ as $s$. Given the initial potential energy as function of height $PE=mgh$ and the kinetic energy as function of velocity $KE=0.5m{v}^2$. The change in height $\mathrm{\Delta }h=h-s$. Here, $m$, $g$ and $h$ are constants and we need to calculate $PE(t)$, $KE(t)$, $PE(h)$, and $KE(h)$.

Calculate Potential Energy as a function of time PE(t)

Use the second equation of motion $$s=ut+0.5g{t}^2$$ As $u=0$, then replacing $s$ in our previous PE formula $$PE(t)=mg(h-0.5g{t}^2)$$

Calculate Kinetic Energy as a function of time KE(t)

We use the first equation of motion $$v=u+gt$$ As $u=0$, the velocity at time $t$ equals $gt$. Then, we can substitute $v$ into the formula for kinetic energy to get $$KE(t)=0.5m(gt{)}^2$$

Calculate Potential Energy as function of Height PE(h)

The gravitational potential energy of an object of mass $m$ at height $h$ is $$PE(h)=mg(h-s)$$. As suggested in the problem statement, the height from which the object falls changes over time. This change in height $\mathrm{\Delta }h=h-s$ can be used to find $PE(h)$ by substituting into the equation for potential energy to get $$PE(h)=mg\mathrm{\Delta }h=mg(h-s)$$

Calculate Kinetic Energy as a function of height KE(h)

The kinetic energy of the falling object depends on its velocity. We can use the velocity equation of motion $$v^2=u^2+2gs$$ As $u=0$, $$v^2=2gs$$. The kinetic energy of the object is given by $$KE(h)=0.5m{v}^2$$. Substituting for velocity, we get $$KE(h)=0.5m(2gs)=mgs$$

Check for conservation of mechanical energy

The total mechanical energy of the system is the sum of kinetic and potential energies. For all times $t$ and heights $h$, the sum of the kinetic and potential energy should be constant, according to the principle of conservation of energy. This can be illustrated on a graph, where the sum of $PE(t)+KE(t)$ and $PE(h)+KE(h)$ would give a straight horizontal line, indicating that the total mechanical energy is constant.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. The formula to calculate kinetic energy (KE) is given by $KE=\frac{1}{2}m{v}^2$. Here, $m$ represents the mass of the object, and $v$ stands for velocity.
When an object is falling from a height, it starts from rest and gains velocity as it falls under the influence of gravity.
This increase in velocity results in an increase in kinetic energy. By using equations of motion, we know that for an object starting from rest, the velocity at time $t$ is $gt$, where $g$ is the acceleration due to gravity. Therefore, the kinetic energy of the object at any time $t$ can be expressed as $KE(t)=\frac{1}{2}m(gt{)}^2$.
Similarly, when considering energy as a function of the height fallen, we find that the kinetic energy at a height $s$, given initial velocity $u=0$, can be determined using $v^2=2gs$. This results in $KE(h)=mgs$, indicating that kinetic energy increases linearly with the distance fallen.
In both cases, as the object falls, its kinetic energy keeps increasing until it reaches maximum just before impact.

Potential Energy

Potential energy is the stored energy of position possessed by an object. For an object at a height, potential energy (PE) is calculated using $PE=mgh$, where $m$ is mass, $g$ is gravitational acceleration, and $h$ is the height above the ground.
As an object falls, its height decreases, leading to a reduction in potential energy. It can be observed as $PE(t)=mg(h-\frac{1}{2}g{t}^2)$ when considering how potential energy changes over time.
This formula indicates that potential energy decreases as the square of time increases, demonstrating how quickly stored energy is converted to kinetic energy.
Similarly, potential energy in terms of height can be expressed as $PE(h)=mg(h-s)$, where $s$ is the distance fallen. This formula shows that potential energy decreases linearly with height.
The significant takeaway here is that all potential energy lost during the fall is transformed into kinetic energy, maintaining total mechanical energy constant.

Equations of Motion

Equations of motion are crucial in calculating how parameters such as velocity, displacement, and time relate to objects in motion. When dealing with objects in free fall under gravity, these equations simplify the calculations.

• First equation of motion: $v=u+gt$
• Second equation of motion: $s=ut+\frac{1}{2}g{t}^2$
• Third equation of motion: $v^2=u^2+2gs$

Since our object starts at rest $(u=0)$, these equations become even more straightforward. The first equation is used to determine velocity at any time $t$.
The second equation helps to determine how far the object has fallen at any particular time, and the third equation is pivotal in relating velocity to height fallen.
All these equations contribute significantly to calculating kinetic and potential energy as the object falls from the initial height to ensure a smooth conversion of potential to kinetic energy.

Graphical Representation

Graphical representation provides a visual comprehension of how energy transforms during the fall of an object. When graphing kinetic energy and potential energy against time or height, one observes clear patterns.
Initially, potential energy is at its maximum and equals total mechanical energy, which is $mgh$. However, as time passes or height decreases, kinetic energy becomes predominant.
The graph showing kinetic energy over time depicts a curve increasing steeply due to velocity squared's contribution. On the other hand, potential energy decreases as time squared, illustrated by a downward curve.
In terms of height, a graph will display a linear decrease in potential energy, while kinetic energy rises linearly as well.
What remains constant throughout is the sum of kinetic and potential energy, reflecting the principle of the conservation of mechanical energy. This can be visualized as a horizontal line on a graph where total mechanical energy remains unchanged in a closed system such as a falling object under gravity.