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The SUVAT Equations of Motion
Kinematics
The SUVAT equations are the five equations that describe motion in a straight line with constant acceleration. They are the most-used tools in kinematics: pick the equation that uses the quantities you know and leaves out the one you do not need, substitute, and solve. Mastering them means you can handle almost any straight-line motion problem.
What SUVAT means
SUVAT stands for the five quantities in the equations: **s** is displacement (m), **u** is initial velocity (m/s), **v** is final velocity (m/s), **a** is acceleration (m/s²), and **t** is time (s). Each equation links four of these five quantities, so if you know any three, you can find a fourth. The equations only work when the acceleration is constant and the motion is in a straight line. For falling objects, a is replaced by g = 9.81 m/s² downward, and for objects slowing down, a is negative.
The five equations
The five equations are: **v = u + at**, **s = ut + ½at²**, **s = ½(u + v)t**, **v² = u² + 2as**, and **s = vt − ½at²**. The first, v = u + at, comes directly from the definition of acceleration. The second adds up the distance covered as the velocity grows. The third uses the average velocity, (u + v) / 2, multiplied by time. The fourth is the most useful when time is not given, and the fifth is the mirror of the second, working backward from the final velocity. You only need to remember the first four well; the fifth follows from them.
Choosing the right equation
The trick to SUVAT problems is choosing the equation that avoids the unknown you do not need. Write down what you know, what you want, and what you do not know and do not need, then pick the equation that contains your knowns and your target but not the unwanted quantity. For example, if time is not given and not asked for, use v² = u² + 2as. If acceleration is not given, use s = ½(u + v)t. Always check the signs: take one direction as positive, give velocities and acceleration the correct signs, and check that your answer has the right size and units.
Worked example: a car accelerating
A car starts at 10 m/s and accelerates at 2 m/s² for 5 s. Find the final velocity: v = u + at = 10 + 2 x 5 = 10 + 10 = 20 m/s. Find the distance: s = ut + 1/2 at² = 10 x 5 + 0.5 x 2 x 5². s = 50 + 25 = 75 m. Check with v² = u² + 2as: 20² = 400, and 10² + 2 x 2 x 75 = 100 + 300 = 400. It matches. Answer: final velocity 20 m/s, distance travelled 75 m.
Common mistakes
- Using SUVAT when acceleration is not constant. Correction: the equations only apply to constant acceleration, for changing acceleration, use graphs or calculus.
- Mixing up displacement and distance. Correction: s is displacement, a vector from start to finish; for motion that reverses direction, distance and displacement differ.
- Forgetting the sign of acceleration. Correction: deceleration is a negative acceleration in the direction of motion, keep the signs consistent throughout.
Practice questions
- A car accelerates from rest at 3 m/s² for 4 s. What is its final velocity?
Show answer
v = u + at = 0 + 3 x 4 = 12 m/s.
- An object moving at 5 m/s accelerates at 2 m/s² over a distance of 12 m. What is its final velocity?
Show answer
v² = u² + 2as = 25 + 2 x 2 x 12 = 25 + 48 = 73, so v = √73 = 8.54 m/s.
- A ball is thrown upward at 20 m/s. How high does it go? Use g = 9.81 m/s².
Show answer
At the top v = 0. 0 = 20² - 2 x 9.81 x s, so s = 400 / 19.62 = 20.39 m.
Frequently asked questions
Which SUVAT equation should I use if I don't know the time?
Use v² = u² + 2as, it is the only one that does not contain t.
What does the S in SUVAT stand for?
Displacement, the straight-line change in position from the start point, measured in metres.