Final velocity ² − initial velocity ² = 2 × acceleration × distance

The idea

The equation v² − u² = 2as relates final velocity v, initial velocity u, uniform acceleration a and distance s. It is provided on the Physics Equations Sheet and is particularly useful when time is not given. For example, if a cyclist speeds up uniformly from 4 m/s to 10 m/s over 21 m: a = (10² − 4²) ÷ (2 × 21) = (100 − 16) ÷ 42 = 2 m/s². Rearranging requires care: v² = u² + 2as, u² = v² − 2as and s = (v² − u²) ÷ (2a). The equation applies only for uniform acceleration in a straight line. Always use standard SI units: m/s for velocity, m/s² for acceleration and m for distance.

What earns the marks

• Select the equation v² − u² = 2as from the Physics Equations Sheet and match each symbol to its quantity.
• Substitute numerical values for v, u, a and s correctly, keeping the subtraction order final velocity squared minus initial velocity squared.
• Rearrange to isolate the required quantity, for example a = (v² − u²) ÷ (2s) or s = (v² − u²) ÷ (2a).
• Use consistent SI units and interpret a negative result for acceleration as deceleration.
• Take the square root when the unknown is v or u, choosing the positive value for speed.

Common mistakes and how to put them right

• Subtracting initial velocity from final velocity before squaring.

  Put it right by squaring each velocity first, then subtracting.

• Dividing by 2 before multiplying by acceleration, or dividing by acceleration before multiplying by 2.

  Put it right by treating 2a as a single product in the denominator.

• Mixing units such as km/h with m/s².

  Put it right by converting all velocities to m/s before substitution.

Exam tips

• Write the rearranged equation before substituting numbers so the structure of the calculation is clear.
• Show the squared values in your working, for example 10² − 4², to make the method visible.