physics

Kinematics Equations

Kinematics equations, often called the SUVAT equations, describe motion in a straight line when acceleration is constant. The main skill is choosing an equation that contains the quantities you know and the one you need, while keeping units and signs consistent.

Visual note, a 3:08 episode with transcript, a mind map, 12 flashcards and a 5 question quiz.

Kinematics Equations at a glance

Visual note summarising kinematics equations

Listen: Kinematics Equations

Two hosts talking it through, 3:08. The full transcript is below, so you can read along or skip the audio.

Transcript

AnnaToday we’re talking about kinematics equations, often called the SUVAT equations. Marco, what does SUVAT stand for?

MarcoThe letters represent five quantities. S is displacement, U is initial velocity, V is final velocity, A is constant acceleration, and T is time. Displacement is measured in metres, velocity in metres per second, acceleration in metres per second squared, and time in seconds.

AnnaWhy is the word constant so important here?

MarcoBecause these equations apply only to straight-line motion with constant acceleration. If acceleration changes, we cannot use them as they are. Also, displacement and velocity can be positive or negative, depending on the direction we choose as positive.

AnnaSo choosing an equation is more than just remembering four formulas. What should we do first?

MarcoFirst, write down what you know. Then identify what you need. Choose an equation containing those quantities, but avoid one containing another unknown. The missing quantity in each equation can help. For example, velocity equals initial velocity plus acceleration multiplied by time does not contain displacement.

AnnaLet’s test that one. A car starts at five metres per second, accelerates at two metres per second squared, and travels for six seconds. What happens?

MarcoWe use final velocity equals initial velocity plus acceleration multiplied by time. That is five plus two multiplied by six, which gives seventeen metres per second. Displacement was not needed.

AnnaWhat if we need displacement, and we know time but not final velocity?

MarcoThen we can use displacement equals initial velocity multiplied by time, plus one half multiplied by acceleration multiplied by time squared. The squared time matters. For a cyclist starting at three metres per second, accelerating at one point five metres per second squared for four seconds, the displacement is twelve plus twelve, or twenty-four metres.

AnnaAnd which equation avoids time altogether?

MarcoFinal velocity squared equals initial velocity squared plus two multiplied by acceleration multiplied by displacement. It is useful when time is unknown or unnecessary. A train starting at ten metres per second, accelerating at two metres per second squared over seventy-five metres, reaches twenty metres per second. We take the physically meaningful square-root result.

AnnaThere is also an equation without acceleration, correct?

MarcoYes. Displacement equals one half multiplied by the sum of initial and final velocity, multiplied by time. The one-half expression is the average velocity for constant acceleration. A runner changing from four to ten metres per second in eight seconds travels fifty-six metres.

AnnaWhat mistakes should listeners watch for?

MarcoDo not confuse distance with displacement, or speed with velocity. Define a positive direction and keep signs consistent, especially when slowing down. Convert to compatible SI units, and never treat an unknown as zero without justification. Most importantly, check that acceleration is constant before using SUVAT.

The notes

What SUVAT means

The letters in SUVAT represent five quantities. s is displacement in metres, u is initial velocity in metres per second, v is final velocity in metres per second, a is constant acceleration in metres per second squared, and t is time in seconds.

Displacement is the change in position, so it can be positive or negative. Velocity is speed with a direction, so it can also be positive or negative. Acceleration describes how velocity changes, and its sign depends on the direction chosen as positive. These equations apply only when acceleration is constant.

How to choose an equation

First, write down the values you know and identify the quantity you need to find. Then choose an equation containing those quantities, but avoid an equation containing another unknown quantity unless you can find that quantity separately.

The equation v = u + at does not contain s. The equation s = ut + 1/2at² does not contain v. The equation v² = u² + 2as does not contain t. The equation s = 1/2(u + v)t does not contain a. This missing quantity can help you choose the best equation.

Equation 1, v = u + at

This equation links initial velocity, final velocity, acceleration and time. It is useful when you know three of these four quantities and need to find the fourth. It is especially useful when displacement is not given or is not needed.

Example: A car starts at 5 m/s and accelerates at 2 m/s² for 6 s. Its final velocity is v = u + at = 5 + (2 × 6) = 17 m/s. Therefore, the car is travelling at 17 m/s after 6 s.

Equation 2, s = ut + 1/2at²

This equation links displacement, initial velocity, acceleration and time. It is useful when time is known and final velocity is not needed. Remember that the acceleration term contains t squared, so calculate the square of the time before multiplying by the other factors.

Example: A cyclist travels for 4 s, starting at 3 m/s and accelerating at 1.5 m/s². The displacement is s = ut + 1/2at² = (3 × 4) + (1/2 × 1.5 × 4²) = 12 + 12 = 24 m. Therefore, the cyclist travels 24 m.

Equation 3, v² = u² + 2as

This equation links initial velocity, final velocity, acceleration and displacement. It is useful when time is not known or is not needed. Since velocities are squared, take care when selecting the physically meaningful answer if you need to square root a result.

Example: A train moving at 10 m/s accelerates at 2 m/s² over 75 m. Its final velocity satisfies v² = u² + 2as = 10² + (2 × 2 × 75) = 100 + 300 = 400. Therefore, v = √400 = 20 m/s.

Equation 4, s = 1/2(u + v)t

This equation links displacement, initial velocity, final velocity and time. It is useful when acceleration is not known or is not needed, provided both the initial and final velocities are known. The expression 1/2(u + v) is the average velocity for constant acceleration.

Example: A runner increases velocity from 4 m/s to 10 m/s in 8 s. The displacement is s = 1/2(u + v)t = 1/2(4 + 10) × 8 = 7 × 8 = 56 m. Therefore, the runner travels 56 m.

What to remember

SUVAT equations apply to straight line motion with constant acceleration.
s means displacement, u means initial velocity, v means final velocity, a means acceleration, and t means time.
Use v = u + at when displacement is not involved.
Use s = ut + 1/2at² when final velocity is not involved and time is known.
Use v² = u² + 2as when time is not involved.
Use s = 1/2(u + v)t when acceleration is not involved and both velocities are known.
Choose a positive direction and use positive or negative signs consistently.
Convert quantities to compatible SI units before substituting into an equation.