Interactive Animation: Mechanical Work – Educational-Animation.org

This animation illustrates the concept of mechanical work \( W = F \cdot s \) using a driving car as an example. Use the “Accelerator pedal position” slider to control the car and observe how the forces (shown as arrows on the car), the work diagram (centre) and the velocity (right) change in real time. In the work diagram, the work done appears as the area beneath the force-distance curve.

What Is Mechanical Work?

In physics, the term “work” has a precise meaning: mechanical work is done whenever a force acts on an object and the object moves in the direction of that force. Work therefore describes the process of energy transfer – it is the link between force and energy.

The basic formula is:

\[ W = F \cdot s \]

Here, \( W \) is the work done (in joules), \( F \) is the applied force (in newtons) and \( s \) is the displacement (in metres). An important detail: only the component of the force that acts along the direction of motion does work. A force perpendicular to the motion – such as gravity when pushing a table horizontally – does not contribute to mechanical work.

Everyday Examples

Mechanical work occurs constantly in everyday life – often without us noticing:

Carrying a suitcase upstairs: Work is done against gravity. The heavier the suitcase and the higher the staircase, the greater the work.
Accelerating a bicycle: The pedalling force accelerates the bicycle – the acceleration work performed is converted into kinetic energy.
Compressing a spring: The force increases with compression. Here, the work equals the area under the force-displacement curve – a typical case for the integral.
Braking a car: The braking work converts kinetic energy into heat. That is why brake discs get hot during hard braking.

In every one of these cases: work = energy transfer. Energy is never lost – it is only converted from one form to another.

Description of the Animation

Use the “Accelerator pedal position” slider to speed up the car. The arrows on the vehicle show the acting forces: the driving force (blue, pointing forward), the total resistance force (red, pointing backward) and the resultant force (green), which produces the acceleration. On the car, the red resistance arrow is also displayed at the tip of the blue driving-force arrow – this makes it immediately visible how much of the driving force is left for acceleration (green arrow) and how much is consumed by resistances.

The middle diagram is the work diagram. It plots force against the distance travelled – the area beneath the curve equals the work done. It is split into three parts:

W_B (green) – acceleration work: it increases the car’s kinetic energy.
W_R (blue) – resistance work: it overcomes rolling and air resistance.
W_Br (red) – braking work: it occurs during full braking.

The right-hand diagram shows the velocity v and the acceleration a plotted against distance. This makes it possible to follow how the car’s motion changes while work is being done.

For a constant force, the work is simply the product of force and distance:

\[ W = F \cdot s \]

If the force changes during the journey, the work is given by the area beneath the force-distance curve:

\[ W = \int F \, ds \]

The acceleration work equals exactly the kinetic energy gained:

\[ W_B = \Delta E_{kin} = \frac{1}{2} \cdot m \cdot v^2 \]

The resistance force is made up of rolling resistance and air resistance:

\[ F_{Roll} = c_r \cdot m \cdot g \qquad F_{Air} = \frac{1}{2} \cdot \rho \cdot c_w \cdot A \cdot v^2 \]

Air resistance grows with the square of the velocity and therefore becomes the dominant resistance at high speeds.

The Three Types of Work in Detail

Acceleration Work (W_B)

When the driving force exceeds the resistance force, the car speeds up. The acceleration work performed is converted entirely into kinetic energy. The more acceleration work is done, the higher the velocity – and because kinetic energy grows with the square of the velocity, accelerating at high speed becomes increasingly costly in terms of energy. You can observe this in the animation: set the accelerator to maximum and watch how the green area (W_B) in the work diagram initially grows quickly but then increases more and more slowly as the rising air resistance claims an ever larger share of the driving force. In the right-hand diagram you can see simultaneously how the acceleration (orange curve) steadily decreases even though the car is still getting faster (blue curve).

Resistance Work (W_R)

Even at constant speed the motor must do work to overcome the driving resistances. This resistance work is converted into heat – through tyre deformation (rolling resistance) and air turbulence (aerodynamic drag). At low speeds rolling resistance dominates; at high speeds air resistance takes over. In the animation this becomes visible once the car reaches its top speed: the green arrow (resultant force) disappears because the entire driving force is consumed by resistances. In the work diagram only the blue area (W_R) continues to grow – no more acceleration work is being done. Also notice how the red resistance arrow is barely visible at low speed but grows steadily longer at high speed – the quadratically increasing air resistance becomes apparent.

Braking Work (W_Br)

During braking, the braking force acts opposite to the direction of motion. Braking work removes kinetic energy from the car and converts it into heat at the brake discs. When the car brakes to a complete stop, the braking work equals exactly the kinetic energy the car had before braking. Press the “Full braking” button in the animation after the car has reached a high speed. The blue driving-force arrow disappears and only the red resistance force remains (braking force plus driving resistances). In the work diagram the red area (W_Br) appears and grows rapidly. In the right-hand diagram you can see the velocity (blue) dropping linearly and the acceleration (orange) jumping to a negative value – characteristic of braking at a nearly constant deceleration.

Understanding the Force-Distance Diagram

The force-distance diagram (work diagram) is one of the most important tools for understanding mechanical work. It plots the applied force \( F \) on the vertical axis against the distance travelled \( s \) on the horizontal axis.

Why Does the Area Under the Curve Equal the Work?

For a constant force the diagram forms a rectangle. The area of that rectangle is height times width = \( F \times s \) – exactly the formula for mechanical work. Select the “constant force” drive model in the animation and set the accelerator to a moderate value. In the work diagram you will indeed see a nearly rectangular area – as long as the speed is still low and air resistance remains small. At higher speeds the blue portion (resistance work) starts to grow while the green portion (acceleration work) shrinks – even though the total height (driving force) stays constant. This illustrates how the motor’s work is distributed among different components. When the force changes during the motion, the path can be divided into many small intervals \( \Delta s \) over which the force is approximately constant. The total work is then the sum of all the small rectangular areas \( F \cdot \Delta s \). In the limit of infinitely small intervals this sum becomes the integral – the exact area under the curve.

What Does the Animation Show in the Work Diagram?

In the animation you can watch the area under the curve grow as the car travels. The three coloured regions show how the total work of the motor is distributed: acceleration work (green), resistance work (blue) and – when braking – braking work (red). The sum of all partial areas always equals the total work done by the motor.

Drive Model

The drive model selector lets you compare two cases:

Constant force: an idealised case – the driving force stays the same regardless of velocity.
Decreasing force: the more realistic model of an electric car. At low speed the force is constant; at higher speed the motor power P limits the force according to \( F = P / v \). The driving force therefore decreases as speed increases.

Constant vs. Variable Force – What Changes?

The animation offers two drive models for comparison. The differences are physically instructive:

With a constant force the driving force stays the same regardless of speed. In the force-distance diagram the driving force appears as a horizontal line. Acceleration decreases as speed rises because air resistance increases and consumes more and more of the constant driving force. Nevertheless, the car can theoretically keep accelerating as long as the driving force exceeds the total resistance.

With a decreasing force (realistic electric-car model) the motor power \( P \) is limited. Since \( P = F \cdot v \), the force must decrease as speed rises. In the force-distance diagram the driving-force curve therefore drops at higher speeds. Top speed is reached sooner and the acceleration phase is shorter.

To see this for yourself, set the accelerator to maximum and switch between the two drive models. In “decreasing force” mode you will see the upper edge of the area in the work diagram falling as distance increases – the force decreases according to \( F = P/v \). In “constant force” mode the upper edge stays horizontal. The right-hand diagram confirms: with decreasing force the velocity curve (blue) flattens earlier and top speed is reached over a shorter distance. Over the same distance the constant-force model does more total work and reaches a higher final speed. The more realistic model, on the other hand, shows why electric cars accelerate superbly from a standstill but lose thrust at high speeds.

Energy Conservation and Energy Conversion

Mechanical work is the central mechanism of energy transfer in mechanics. It links force to energy: wherever work is done, energy is converted from one form into another.

The driving car illustrates the energy conversions step by step:

Acceleration: The motor does work – chemical energy (battery) is converted into kinetic energy.
Driving against resistances: The motor does work – energy is converted into heat (tyres, air).
Braking: The brake does (negative) work – kinetic energy is converted into heat.

The law of energy conservation states that the total energy in a closed system remains constant. No energy is lost – it is only converted. In the animation this can be read directly: accelerate the car and then brake to a complete stop. In the work diagram you can see the three components of the motor’s work: the green area (W_B) shows the energy converted into kinetic energy, the blue area (W_R) the energy lost to driving resistances as heat, and the red area (W_Br) the energy converted into heat during braking. The sum of all areas equals the total work done by the motor.

This principle applies not just to cars but to every mechanical system – from a falling stone to a swinging spring to a roller coaster.

Summary: Key Formulas

Formula	Meaning	Applies to
\( W = F \cdot s \)	Mechanical work	Constant force along the direction of motion
\( W = \int F \, ds \)	Mechanical work (general)	Variable force
\( W_B = \frac{1}{2} m v^2 \)	Acceleration work	Acceleration from rest
\( F_{Roll} = c_r \cdot m \cdot g \)	Rolling resistance	All speeds
\( F_{Air} = \frac{1}{2} \rho \, c_w A \, v^2 \)	Air resistance	Speed-dependent
\( F = P / v \)	Force at limited power	Electric-car model
\( E_{kin} = \frac{1}{2} m v^2 \)	Kinetic energy	Any moving object