Energy is the ability to do work, and in mechanics it comes in two main forms: kinetic energy, the energy of motion, and potential energy, the energy stored by position. A roller coaster at the top of a hill has plenty of the second kind. Halfway down it has some of each, and at the bottom it is almost all the first.
This article explains both forms, the rule that connects them, and what changes when friction enters. A pendulum simulator lets you watch the energy move between the bars as the bob swings.
Kinetic energy
Any object with mass $latex m$ moving at speed $latex v$ has kinetic energy:
Kinetic energy
$latex KE = \dfrac{1}{2} \times m \times v^2$
The unit is the joule (J): 1 J = 1 kg·m²/s². Because the speed is squared, doubling the speed multiplies the kinetic energy by four. That is why a crash at 100 km/h is far worse than one at 50 km/h, and why stopping distances grow so quickly with speed.
| Speed of a 1000 kg car | Kinetic energy |
|---|---|
| 10 m/s (36 km/h) | 50 kJ |
| 20 m/s (72 km/h) | 200 kJ |
| 30 m/s (108 km/h) | 450 kJ |
Gravitational potential energy
An object at height $latex h$ above a chosen reference level stores energy because gravity can pull it down:
Gravitational potential energy
$latex PE = m \times g \times h$
Use $latex g = 9.8\ \text{m/s}^2$. The reference level, where $latex h = 0$, is your choice: the floor, the table or the lowest point of the motion. Only changes in potential energy matter, so any level works as long as you keep it fixed through the problem. Springs store elastic potential energy too, $latex \tfrac{1}{2} k x^2$, but this article stays with gravity.
Conservation of mechanical energy
The mechanical energy is the sum $latex E = KE + PE$. When only gravity does work on the object (no friction, no air drag), the sum stays constant:
$$\begin{gathered}\frac{1}{2} m v_1^2 + m g h_1 \\ = \frac{1}{2} m v_2^2 + m g h_2\end{gathered}$$
The mass appears in every term, so it cancels. The speed at a lower point only depends on the height drop:
Speed after dropping a height
$latex v = \sqrt{2 \times g \times \Delta h}$
This is the same result as in free fall, obtained without any time or acceleration. It works for curved paths too, since gravity's work depends only on the height change. The figure shows a ball falling from 10 m: the potential part shrinks exactly as much as the kinetic part grows.
Pendulum with live energy bars
Release the bob and watch. At the highest points the pendulum stops for an instant, so all the energy is potential. At the lowest point the height is zero and all the energy is kinetic. The total stays flat. Switch on air resistance and the total bar shrinks while the heat bar grows.
Pendulum with live energy bars
What changes when there is friction
Energy is never destroyed, but mechanical energy can turn into thermal energy through friction or air drag. The accounting is:
$$E_{initial} = E_{final} + \text{energy lost as heat}$$
The heat you cannot easily recover is what makes real bounces smaller and real swings fade. When a problem gives you the final speed and asks for the energy lost, compare the initial and final mechanical energy and attribute the difference to friction.
Solved examples
Example 1
A 1200 kg car travels at 20 m/s. Find its kinetic energy. What happens if the speed doubles?
$$KE = \frac{1}{2} \times 1200 \times 20^2 = 240000\ \text{J}$$
The car has 240 kJ. At 40 m/s the square makes the energy 960 kJ, which is four times larger.
Example 2
A 0.5 kg ball is dropped from a height of 10 m. Find its speed just before it hits the ground and its kinetic energy there.
Energy conservation between the top and the ground gives:
$$v = \sqrt{2 g h} = \sqrt{2 \times 9.8 \times 10} = 14\ \text{m/s}$$
$$KE = \frac{1}{2} \times 0.5 \times 14^2 = 49\ \text{J}$$
Check: the potential energy at the top was $latex 0.5 \times 9.8 \times 10 = 49$ J. The ball arrives at 14 m/s with 49 J.
Example 3
A pendulum of length 1.5 m is released from rest at 60° from the vertical. What is the speed of the bob at the lowest point?
The bob rises above its lowest point by $latex h = L(1 - \cos\theta)$:
$$h = 1.5 \times (1 - \cos 60^{\circ}) = 1.5 \times 0.5 = 0.75\ \text{m}$$
$$v = \sqrt{2 \times 9.8 \times 0.75} = \sqrt{14.7} = 3.83\ \text{m/s}$$
The bob passes the lowest point at 3.83 m/s, whatever its mass. Set the lab to 60° and the position slider to the middle to see the energy turn fully kinetic.
Example 4
A 2 kg block slides from rest down a ramp 3 m high and reaches the bottom at 6 m/s. How much energy was lost to friction?
$$E_{initial} = m g h = 2 \times 9.8 \times 3 = 58.8\ \text{J}$$
$$E_{final} = \frac{1}{2} m v^2 = \frac{1}{2} \times 2 \times 6^2 = 36\ \text{J}$$
The difference went into heat: $latex 58.8 - 36 = 22.8$ J. About 39% of the initial energy was lost.
Common mistakes
- Forgetting to square the speed. Kinetic energy depends on $latex v^2$, not on $latex v$.
- Using grams or km/h directly. Convert to kg and m/s first, or the answer will not be in joules.
- Changing the reference level midway. Pick the zero of height once and keep it.
- Using conservation of mechanical energy with friction. With friction, mechanical energy decreases. Account for the heat or use the full energy balance.
- Thinking energy is used up. Total energy is conserved. Mechanical energy turns into other forms, mostly heat.
Practice problems
Problem 1
A baseball of mass 0.145 kg moves at 40 m/s. What is its kinetic energy?
Answer
$$KE = \frac{1}{2} \times 0.145 \times 40^2 = 116\ \text{J}$$
The kinetic energy is 116 J.
Problem 2
A 70 kg climber is 15 m above the ground. Find the gravitational potential energy relative to the ground.
Answer
$$PE = 70 \times 9.8 \times 15 = 10290\ \text{J}$$
That is 10.29 kJ.
Problem 3
A roller coaster car starts from rest at the top of a 30 m hill. Ignoring friction, what is its speed at 10 m above the ground?
Answer
$$v = \sqrt{2 \times 9.8 \times (30 - 10)} = \sqrt{392} = 19.8\ \text{m/s}$$
The speed is 19.8 m/s, about 71 km/h.
Problem 4
A skateboarder starts from rest at the top of a frictionless ramp 1.2 m high. How fast is the skateboarder at the bottom?
Answer
$$v = \sqrt{2 \times 9.8 \times 1.2} = \sqrt{23.52} = 4.85\ \text{m/s}$$
The speed is 4.85 m/s, independent of the skateboarder's mass.
Problem 5
A 3 kg object is thrown straight up at 12 m/s. Use energy to find the maximum height.
Answer
At the top all the kinetic energy has become potential: $latex \tfrac{1}{2} m v^2 = m g h$.
$$h = \frac{v^2}{2 g} = \frac{12^2}{2 \times 9.8} = 7.35\ \text{m}$$
The maximum height is 7.35 m. The mass cancels.
Problem 6
A 0.2 kg ball is dropped from 5 m and rebounds to 3.5 m. How much mechanical energy is lost in the bounce?
Answer
$$\Delta E = m g (h_1 - h_2) = 0.2 \times 9.8 \times (5 - 3.5) = 2.94\ \text{J}$$
The ball loses 2.94 J, which is 30% of its initial energy.
Frequently asked questions
What is the difference between kinetic and potential energy?
Kinetic energy depends on motion: mass and speed. Potential energy depends on position or configuration: mass and height for gravity. Objects often trade one for the other.
Can energy be negative?
Kinetic energy never is, since $latex v^2$ is positive. Potential energy can be negative when the object is below the chosen reference level, which is fine because only differences matter.
Why does the mass cancel in the speed of a falling object?
Both the kinetic energy and the potential energy are proportional to the mass. A heavier object has more energy to convert, but it also has more mass to speed up, and the two effects cancel.
Is mechanical energy always conserved?
Only when no friction, drag or other non-conservative force acts. In real life some mechanical energy becomes heat, and the total energy, counting heat, is what stays constant.
Related calculators
- Kinetic and potential energy calculator: compute KE or PE from mass, speed and height.
- Free fall calculator: speed and time for a falling object.
- km/h to m/s calculator: convert speeds before using the energy formulas.
See also
- Solved problems on free fall: the same speeds from the kinematic point of view.
- Vertical projectile motion: maximum height and time of flight.
- Acceleration due to gravity: the value of g and how it varies.
- Kinematic equations for constant acceleration: compare with the energy method.