Kinetic and potential energies (3.3.2)
On this page
Kinetic energy is the energy possessed by an object due to its motion. Kinetic energy is a scalar quantity that has a magnitude but no direction. The formula that describes the kinetic energy is:
Where:
- is the mass in kilograms (), and
- is the speed in metres per second ().
The kinetic energy of an object is measured in joules () and is directly proportional to its mass and to the square of its speed.
For example, doubling the mass of an object moving at the same speed would double its kinetic energy. Alternatively, doubling the velocity of the same object would quadruple its kinetic energy.
To derive the kinetic energy expression from the first principles, we use the following equations:
- Work expression:
- Newton’s second law:
- From the equations of motion:
Where and both symbolise displacement in different contexts.
Replacing and rearranging the previous equations, we get:
For , the previous expression becomes:
Question walkthrough
Deriving the Kinetic Energy Formula
Derives KE = ½mv² from first principles by combining the work-energy principle, Newton's second law, and the SUVAT equations for a car accelerating from rest.
Gravitational potential energy (GPE) is the energy possessed by an object due to its position in a gravitational field. GPE is a scalar quantity that has a magnitude but no direction:
Where:
- is the mass in kilograms (),
- is the height above an arbitrary reference level in metres (), and
- is the gravitational field strength in metres per second squared ().
The gravitational potential energy of an object is measured in joules (). It can be positive, negative, or zero.
In the diagram above a ball at:
- the zero reference has no gravitational potential energy,
- a ball at a distance above the reference has a gravitational potential energy equal to , and
- a ball at a distance below the reference has a gravitational potential energy equal to .
Since gravitational potential energy is defined from a reference point, it is often required to determine the change in the gravitational potential energy between two positions.
The diagram below shows a ball thrown upward. The ball starts from A, rises to its maximum height at B, then falls back towards its initial vertical position at C. Taking the zero reference at the level of the hand:
- The gravitational potential energy at A and C is equal to zero
- The gravitational potential energy at B is equal to
The change in gravitational potential energy from point A to point B:
Now the change in gravitational potential energy from point B to point C:
This implies that any rising object gains gravitational potential energy while any falling object loses gravitational potential energy.
The expression for gravitational potential energy can be derived from first principles, using the fact that the work done in lifting an object is stored as a change in gravitational potential energy.
A box is lifted from the ground to a height as shown below. In the process, a lifting force equal to the box’s weight is applied.
The work done by the lifting force against gravity is stored as a change in gravitational potential energy of the box :
The expression for the work done by a force moving an object a distance is:
Therefore, the previous expression can be rewritten as:
The lifting force equals the box’s weight:
Since the box is initially at the zero reference, the is equal to zero. Substituting the expression for and removing :
Question walkthrough
GPE Change Independent of Zero Reference
Calculates gravitational potential energy at three points using two different zero references, showing that the change in GPE between any two points is the same regardless of the reference chosen.
The exchange between kinetic energy and gravitational potential energy occurs for an object moving in a gravitational field. Examples include:
- An object that falls from the roof of a building loses gravitational potential energy and gains kinetic energy.
- A swing at its highest position, moving towards its lowest position, loses gravitational potential energy while gaining kinetic energy.
- When a ball is thrown up with an initial velocity, it loses kinetic energy as it ascends and gains gravitational potential energy.
For an object moving in a gravitational field, in the absence of frictional forces, the gain in one form of energy is equal to the loss in the other:
If a ball at A possesses of gravitational potential energy and falls, it loses of by the time it reaches B. This loss in is converted to kinetic energy. Therefore, the ball possesses of when it reaches the ground.
Although systems in a gravitational field may look different, the way that their energy changes between kinetic and gravitational in the absence of frictional forces is the same.
General statements can be made about objects in a gravitational field released from a raised level or projected upwards from a lower level:
- At maximum height, an object’s kinetic energy is minimal, and its gravitational potential energy is at its maximum.
- At ground level, an object’s gravitational potential energy is minimal, and its kinetic energy is at its maximum.
- At intermediate heights, kinetic energy and gravitational potential energy will vary, but their sum remains constant, which can be written as:
This is a statement of the law of energy conservation.
Question walkthrough
Energy Conservation on a Frictionless Slope
Uses GPE = mgh and KE = ½mv² to track energy conversion for a frictionless skier's descent, then finds the resulting speed at the bottom of the slope.