Changes in energy: Students should be able to calculate the amount of energy associated with a moving object, a stretched spring and an object raised above ground level.

The idea

Three equations cover the energy stores named in this statement. For a moving object, kinetic energy E_k = ½mv², where m is mass in kilograms and v is speed in metres per second. For a stretched spring, elastic potential energy E_e = ½ke², where k is the spring constant in newtons per metre and e is the extension in metres. For an object raised above ground level, gravitational potential energy E_p = mgh, where m is mass in kilograms, g is gravitational field strength in newtons per kilogram and h is height in metres.

For example: A 2 kg object moving at 3 m/s has E_k = ½ × 2 × 3² = 9 J. A spring with k = 200 N/m stretched by 0.1 m has E_e = ½ × 200 × 0.1² = 1 J. A 2 kg object raised 5 m with g = 9.8 N/kg has E_p = 2 × 9.8 × 5 = 98 J. Always square the speed or extension before multiplying by the other terms.

What earns the marks

• Selects the correct equation for the store described: E_k = ½mv² for a moving object, E_e = ½ke² for a stretched spring, or E_p = mgh for an object raised above ground level.
• Substitutes values with correct units, converting grams to kilograms, centimetres to metres and kilometres per hour to metres per second where needed.
• Squares the speed or the extension before completing the multiplication, for example 3² = 9 in E_k = ½ × 2 × 3².
• Gives the answer in joules with an appropriate number of significant figures and states the unit.
• Rearranges an equation to find mass, speed, spring constant, extension or height when the energy is given.

Common mistakes and how to put them right

• Forgetting to square the speed or extension.

  Put it right: Write the squared term first, for example v² = 3² = 9, before multiplying.

• Using the total length of a spring instead of its extension.

  Put it right: Extension = stretched length − original length.

• Using grams or centimetres directly in the equation.

  Put it right: Convert to kilograms and metres before substituting.

Exam tips

• Write the equation, then the substitution with units, then the answer with its unit, so method marks are visible.
• Check whether the question gives diameter, total length or extension, and use only the extension in E_e = ½ke².
• For 'show that' questions, work to more significant figures than the given answer and round only at the end.