SUVAT Equation: Displacement (initial velocity and time)

Core idea

Overview

This equation represents the area under a velocity-time graph, where the 'ut' term accounts for the rectangular area of initial velocity and the '0.5at²' term accounts for the triangular area resulting from acceleration. It is a fundamental kinematic relation that assumes the acceleration remains uniform throughout the entire duration of motion.

When to use: Use this formula when you know the initial velocity, constant acceleration, and the time elapsed, but do not know the final velocity.

Why it matters: It is essential for predicting the exact position of moving objects, such as vehicles braking to a stop or projectiles in flight, which is critical in engineering and transport safety.

Symbols

Variables

s = Displacement, u = Initial Velocity, a = Acceleration, t = Time

s

Displacement

Variable

u

Initial Velocity

Variable

a

Acceleration

Variable

t

Time

Variable

Walkthrough

Derivation

Derivation of SUVAT Equation: Displacement (initial velocity and time)

This equation is derived by calculating the area under a velocity-time graph for an object undergoing constant acceleration. It represents the total displacement as the sum of the initial velocity component and the change in velocity component.

The motion occurs in a straight line
The acceleration (a) is constant throughout the time interval

1

Analyze the Velocity-Time Graph

We begin with the definition of constant acceleration, where the final velocity (v) is the initial velocity (u) plus the product of acceleration (a) and time (t).

v = u + a t

Note: The area under a v-t graph equals displacement.

2

Define Displacement as the Area

On a velocity-time graph, the displacement (s) is the area under the line. This area consists of a rectangle (base t, height u) and a right-angled triangle (base t, height at).

s = Area of Rectangle + Area of Triangle

Note: The height of the triangle is (v - u), which equals at.

3

Calculate the Areas

We substitute the geometric formulas for the area of the rectangle (base ×height) and the triangle (1/2 ×base ×height) using the variables from the graph.

s = (u \times t) + \frac{1}{2} (t \times a t)

Note: Ensure units are consistent throughout the calculation.

4

Simplify the Equation

By multiplying the terms in the second part of the equation, we arrive at the final SUVAT expression.

s = u t + \frac{1}{2} a t^{2}

Note: This is often written as s = ut + 0.5at^2.

Result

s = u t + \frac{1}{2} a t^{2}

Source: AQA Physics Specification (7408) / OCR Physics A (H556)

Free formulas

Rearrangements

Solve for $u$

Make u the subject

u = \frac{s - 0.5 a t ^{2}}{t}

Isolate the term containing u by subtracting the acceleration component and dividing by time.

Difficulty: 2/5

Solve for $a$

Make a the subject

a = \frac{2 ( s - u t )}{t ^{2}}

Isolate the acceleration term by moving initial velocity and then multiplying by the reciprocal of time squared.

Difficulty: 3/5

Solve for $t$

Make t the subject

t = \frac{- u \pm u ^{2} + 2 a s}{a}

Rearrange as a quadratic equation in terms of t and solve using the quadratic formula.

Difficulty: 5/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Think of this as calculating the area under a velocity-time graph. A constant acceleration creates a trapezoid: the 'ut' term is the rectangular base representing the distance covered at a steady initial speed, while the '0.5at²' term is the triangular area on top representing the extra distance gained due to the gradual increase in speed.

s

Displacement

The total net change in position from the starting point.

u

Initial velocity

How fast the object is moving at the exact moment you start your stopwatch.

t

Time

The duration of the interval over which the motion is being observed.

a

Acceleration

The rate at which the velocity is changing; how quickly the object is speeding up or slowing down.

Signs and relationships

0.5: Derived from the area of a triangle formula (1/2 * base * height); it accounts for the fact that the object gains speed linearly rather than instantly.
+: Indicates that the 'extra' distance gained by accelerating adds to the base distance covered by the initial velocity.
a: If acceleration is in the opposite direction to the initial velocity (deceleration), 'a' must be assigned a negative sign to reflect the loss of displacement.

One free problem

Practice Problem

A cyclist starts from rest and accelerates at 2 m/s² for 5 seconds. How far has the cyclist traveled?

Initial Velocity0

Acceleration2

Time5

Solve for: $s$

Hint: Since the cyclist starts from rest, u = 0, so the equation simplifies to s = 0.5 * a * $t^{2}$ .

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In how far a car will travel while accelerating from a standstill at traffic lights to reach a specific speed within a certain timeframe, SUVAT Equation: Displacement (initial velocity and time) is used to calculate Displacement from Initial Velocity, Acceleration, and Time. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Ensure all units are consistent (e.g., meters, seconds) before substituting values.
Remember that displacement is a vector; direction matters, so define a positive direction and stick to it.
If an object starts from rest, the initial velocity 'u' is zero, simplifying the calculation to s = 0.5at².

Avoid these traps

Common Mistakes

Forgetting to square the time variable (t²).
Confusing displacement (s) with total distance traveled if the object changes direction.
Applying this to situations where acceleration is not constant.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This equation is derived by calculating the area under a velocity-time graph for an object undergoing constant acceleration. It represents the total displacement as the sum of the initial velocity component and the change in velocity component.

Use this formula when you know the initial velocity, constant acceleration, and the time elapsed, but do not know the final velocity.

It is essential for predicting the exact position of moving objects, such as vehicles braking to a stop or projectiles in flight, which is critical in engineering and transport safety.

Forgetting to square the time variable (t²). Confusing displacement (s) with total distance traveled if the object changes direction. Applying this to situations where acceleration is not constant.

In how far a car will travel while accelerating from a standstill at traffic lights to reach a specific speed within a certain timeframe, SUVAT Equation: Displacement (initial velocity and time) is used to calculate Displacement from Initial Velocity, Acceleration, and Time. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Ensure all units are consistent (e.g., meters, seconds) before substituting values. Remember that displacement is a vector; direction matters, so define a positive direction and stick to it. If an object starts from rest, the initial velocity 'u' is zero, simplifying the calculation to s = 0.5at².

References

Sources

Young and Freedman, University Physics with Modern Physics
A-Level Physics: Edexcel/AQA Specification Guides
AQA Physics Specification (7408) / OCR Physics A (H556)

Overview

Variables

Derivation

Analyze the Velocity-Time Graph

Define Displacement as the Area

Calculate the Areas

Simplify the Equation

Rearrangements

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

SUVAT Equation: Final Velocity

Frequently Asked Questions

Sources