Mechanics

1D Kinematics

Start

Learn position, velocity, and acceleration through graph slopes and a stop-in-zones challenge.

2D Kinematics

Extend kinematics into vector components, projectile motion, and two-dimensional acceleration.

Forces

Explore common mechanics forces with Newt the physics frog, from spring-like contact forces to gravity and friction.

Collisions

Compare elastic and inelastic collisions while keeping momentum and energy bookkeeping straight.

Energy

Here

Build a 3D roller coaster and watch kinetic, potential, and thermal energy trade as the train runs.

Energy

Energy is one of the most useful ideas in physics because it is a single number that we can track as a system changes. In mechanics we will meet it in two forms first: the energy an object has because it is moving, and the energy it has because of where it sits in a gravitational field. This page introduces both, then lets you watch one turn into the other by picking up a ball and dropping it.

Kinetic Energy

Kinetic energy is the energy of motion. For an object of mass $m$ moving with speed $v$ , it is defined as

K = \tfrac{1}{2}mv^2.

Two features of this definition are worth pausing on. First, kinetic energy grows with mass: a heavier object moving at the same speed carries more of it. Second, and more importantly, it depends on the square of the speed. Doubling the speed does not double the kinetic energy — it quadruples it. Kinetic energy is measured in joules (J), where $1\,\text{J} = 1\,\text{kg}\cdot\text{m}^2/\text{s}^2$ .

Gravitational Potential Energy

Gravitational potential energy is energy stored by virtue of an object’s height. Near Earth’s surface, where the gravitational field strength $g \approx 9.8\,\text{m/s}^2$ is essentially constant, it is defined as

U = mgh,

where $h$ is the height of the object above a chosen reference level. Lifting a ball higher stores more potential energy; letting it down spends that store. Because only changes in height matter, the choice of where $h = 0$ is just bookkeeping. It is common to measure height from ground level.

Potential energy is also measured in joules, which is what makes the next step possible: because $K$ and $U$ are measured in the same units, we can add them and watch the total as an object moves.

Energy Conservation

Drag the ball up to store gravitational potential energy, then let go. As it falls, its height drops and its speed grows — potential energy is steadily converted into kinetic energy. This illustrates conservation of mechanical energy, a special case of the more general conservation principle.

Mass (kg)2.0

Kinetic0 J

Potential0 J

height = 0.00mspeed = 0.00m/stotal energy = 0J

The diagram on the right is a convenient way to visualize the energy distribution of the system. Each colored block is one form of energy, drawn to scale. The purple block is gravitational potential energy, the green block is kinetic energy, and they are stacked so the height of the whole column is the total mechanical energy.

Worked Examples

Problem Solving

Speed of a Dropped Ball

Problem

A $2.0\,\text{kg}$ ball is held at rest $3.0\,\text{m}$ above the ground and released. How fast is it moving just before it lands?

Given

$m = 2.0\,\text{kg}$
$h = 3.0\,\text{m}$
released from rest, so $K_{\text{top}} = 0$

Use

At the top, all of the energy is potential:

U_{\text{top}} = mgh = (2.0)(9.8)(3.0) = 58.8\,\text{J}.

By the time it reaches the ground, all of that has become kinetic energy, so $K_{\text{bottom}} = 58.8\,\text{J}$ . Setting $\tfrac{1}{2}mv^2 = K_{\text{bottom}}$ and solving for the speed:

v = \sqrt{\frac{2K_{\text{bottom}}}{m}} = \sqrt{\frac{2(58.8)}{2.0}}.

Solve

v = \sqrt{58.8} \approx 7.7\,\text{m/s}.

Check

Notice the mass cancels: $v = \sqrt{2gh}$ . Two balls of different mass dropped from the same height land at the same speed — exactly what the explorer shows when you change the mass slider.

Reading the Bar Chart Partway Down

Problem

A $1.5\,\text{kg}$ ball is dropped from $4.0\,\text{m}$ . When it has fallen to a height of $1.0\,\text{m}$ , how much energy is kinetic and how much is potential?

Given

$m = 1.5\,\text{kg}$
release height $h_0 = 4.0\,\text{m}$ , current height $h = 1.0\,\text{m}$

Use

The total energy equals the potential energy at the release point:

E = mgh_0 = (1.5)(9.8)(4.0) = 58.8\,\text{J}.

The potential energy now is

U = mgh = (1.5)(9.8)(1.0) = 14.7\,\text{J},

and the rest of the total must be kinetic.

Solve

K = E - U = 58.8 - 14.7 = 44.1\,\text{J}.

Check

On a stacked bar chart, draw a purple block of $14.7\,\text{J}$ with a green block of $44.1\,\text{J}$ on top — together they reach the same $58.8\,\text{J}$ total line as at the start.

Energy Checkpoint

Question 1 of 3

Two carts have the same mass, but cart B moves twice as fast as cart A. How do their kinetic energies compare?

Choose an answer to get instant feedback.

Take It Further: Build a Coaster

The same trade between kinetic and gravitational potential energy plays out in a roller coaster: every drop converts potential energy into kinetic, and every uphill converts it back. Build a track, release the train, and open Telemetry to track the mechanical energy.

Coaster Builder

Stack track pieces, release the train, and watch kinetic and potential energy trade as it runs. Scroll to zoom, right-drag to pan, WASD to move the camera.