# SUVAT Calculator

Our SUVAT calculator can help you calculate the kinematic *equations of motion* when the body is * accelerating uniformly*. This article contains everything you need to know about SUVAT formulae, so grab some snacks and join us below for a brief discussion about:

- What does SUVAT stand for?
- Deriving SUVAT equations
- SUVAT problems

## What does SUVAT stand for?

Kinematics is the study of motion without looking at the forces that cause this motion. In this field, the **SUVAT equations** calculate the linear motion of bodies under constant acceleration. They describe the relationship between the body's

*displacement*$s$,*initial velocity*$u$,*final velocity*$v$,*uniform acceleration*$a$, and*time*$t$.

Now can you guess what SUVAT stands for?

There are five SUVAT equations, and we shall derive them individually.

## SUVAT formulae for velocities

To describe the initial and final velocities of a body under constant acceleration, consider the following velocity-time graph:

The *gradient* of this graph is equal to the *acceleration* $a$. Mathematically:

If the motion starts at $t = 0$ and ends at time $t = t$, we can write:

Rearranging this, we get the **first SUVAT equation**:

## SUVAT formula for displacement

The area under the velocity-time graph in Figure 1 is the displacement of the body. Notice that this area is a trapezoid. The formula for the area of a trapezoid is:

Looking at the graph from the right side, the base lengths (parallel sides) are $u$ and $v$, and the height is equal to $t$. Thus we arrive at the **second SUVAT equation**:

Substituting the first SUVAT equation here:

This is the **third SUVAT equation**. Notice that we can also substitute the relation for initial velocity instead and get the **fourth SUVAT equation:**

## SUVAT formulae without time

All three SUVAT equations so far involve the variable time $t$. But what if we don't know the time? So we derive the following SUVAT equation by eliminating $t$.

Rearranging the first SUVAT equation gives:

Substituting this into the trapezoid area formula:

Let's summarise all five SUVAT equations here:

S.no | SUVAT equation |
---|---|

1 | $v = u + at$ |

2 | $s = \frac{(u + v)}{2} \cdot t$ |

3 | $s = ut + \frac{1}{2}at^2$ |

4 | $s = vt - \frac{1}{2} at^2$ |

5 | $v^2 = u^2 + 2as$ |

## SUVAT problems for practice

Let's go through a couple of SUVAT problems for practice.

- A stone dropped from
**50 m high**freely falls to the ground under**gravitational acceleration**. With what*velocity*does it reach the ground, and*how long*does it take?

What we know:

- Height (or displacement) $s = 50 \text{ m}$.
- Initial velocity $u = 0$.
- Acceleration $a = g = 9.81 \text{ m/s}^2$, if we define positive direction is downwards.

To find:

- Final velocity $v$.
- Time $t$.

Using the fifth SUVAT equation:

Substituting this in the first SUVAT equation:

- A squirrel is running away from an unleashed dog. Within 5 seconds, it safely climbs up a tree 20 m away. Both the squirrel and the dog started from rest, and the dog is initially 5 m away from the squirrel. If the squirrel barely made it in time, find out the acceleration of these two animals.

What we know about the squirrel:

- Displacement $s_s = 20 \text{ m}$.
- Initial velocity $u_s = 0$.
- Time $t_s = 5 \text{ s}$.

What we know about the dog:

- Displacement $s_d = 25 \text{ m}$.
- Initial velocity $u_d = 0$.
- Time $t_d = 5 \text{ s}$.

To find:

- Acceleration of the two animals.

Rearranging the third SUVAT equation to find acceleration:

For the squirrel:

For the dog:

## How to use this SUVAT calculator

This SUVAT calculator is a powerful tool that can calculate all SUVAT equations.

- Enter any
among**three known parameters***displacement, initial and final velocities, acceleration, and time*, and the calculator will determine the remaining two. - You cannot modify the two calculated values without
**resetting**the whole calculator. Click on the reset button at the bottom left of the calculator.

Interested in more kinematics? Try our projectile motion calculator!