Kinematics Equations (SUVAT): How to Pick the Right One

What do the SUVAT symbols mean?

All except time are vectors, so you must choose a positive direction and keep to it. Slowing down means a negative acceleration (or a positive one if you chose the opposite direction as positive).

The four constant acceleration equations

Each of the constant acceleration equations leaves out one variable:

Textbooks and AP Physics 1 write the same equations with different letters: v = v₀ + at, x = x₀ + v₀t + ½at² and v² = v₀² + 2a(x − x₀). If your course has an equation sheet, check it with the AP Physics 1 equation sheet.

How to choose the right equation

1. List the five variables and write down the three you know.
2. Identify the unknown you want to find.
3. Pick the equation that leaves out the variable you neither know nor want.
4. Rearrange, substitute with units and solve. Check that the answer is sensible.

For example, if you know u, a and s, and want v, then t is the missing variable, so use v² = u² + 2as.

When can you use these equations?

Only when acceleration is constant, and the motion is in a straight line. They work for a car braking steadily, an object falling near Earth's surface (with a = g = 9.8 m/s² and no air resistance) and a ball thrown straight up. They do not work if the acceleration changes, as it does for a car speeding up and slowing down in traffic. In that case use graphs instead, as in motion graphs.

Example 1: a car accelerating from rest

A car starts from rest and accelerates at 3 m/s² for 5 s. Find its final speed and the distance covered.

• Known: u = 0, a = 3 m/s², t = 5 s.
• Final speed: v = u + at = 0 + 3 × 5 = 15 m/s.
• Distance: s = ut + ½at² = 0 + ½ × 3 × 5² = 37.5 m.

Example 2: braking distance

A car travelling at 20 m/s brakes with a constant deceleration of 5 m/s². How far does it travel before stopping?

• Known: u = 20 m/s, v = 0, a = −5 m/s². Time is not given.
• Use v² = u² + 2as: 0 = 20² + 2 × (−5) × s, so 10s = 400.
• s = 40 m.

The time can then be found from v = u + at: 0 = 20 − 5t, so t = 4 s. Doubling the speed to 40 m/s would make the stopping distance four times as long (160 m), because s depends on u².

Example 3: free fall

A ball is dropped from a height of 20 m. How long does it take to hit the ground, and how fast is it moving?

• Known: u = 0, a = 9.8 m/s² (down is positive), s = 20 m.
• Time: s = ut + ½at² gives 20 = 4.9t², so t² = 4.08 and t = 2.0 s.
• Speed: v = u + at = 9.8 × 2.02 = 19.8 m/s, or about 20 m/s.

For objects that are thrown at an angle, apply the same equations separately to the horizontal and vertical motion, as in the best angle for maximum range.

Common mistakes

• Using the equations when acceleration is not constant.
• Mixing units, such as km/h with m/s. Convert everything to metres and seconds first.
• Forgetting signs. If up is positive, then g is −9.8 m/s² for a ball thrown upwards.
• Confusing distance and displacement. s is displacement, which can be zero if you return to the start.
• Using an average speed as u or v.

Frequently asked questions

What are the four kinematics equations?

They are v = u + at, s = ut + ½at², v² = u² + 2as and s = ½(u + v)t. They link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t) for motion with constant acceleration.

What does SUVAT stand for?

SUVAT is the list of variables: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time. The name reminds you that each equation uses four of the five.

What value of acceleration should I use for free fall?

Use g = 9.8 m/s² (or 9.81) downwards near Earth's surface, ignoring air resistance. Some courses round it to 10 m/s² for quick work. If you choose up as positive, then a = −9.8 m/s².

Can I use the kinematics equations for circular motion?

No. They assume straight-line motion with constant acceleration. In circular motion the direction of the velocity changes continuously, so different equations are needed.

Try it yourself

Open Motion Graphs, set an initial velocity and an acceleration, and read off the velocity at t = 5 s. Then check your reading with v = u + at. Next, try the area under the velocity–time graph and compare it with s = ut + ½at².