# Constant Acceleration Equations

## The Versatility of SUVAT Equations in Solving for Motion Variables

In physics, the SUVAT equations play a crucial role in connecting and solving for five different variables of motion. These variables are displacement, initial velocity, final velocity, acceleration, and time taken.

## Mastering the 5 Constant Acceleration Equations

To excel in solving motion problems, it is essential to be well-versed in the five SUVAT equations. Each equation serves a unique purpose in connecting the motion variables.

• v = u + at
• s = ½(u + v)t
• s = ut + ½at²
• s = vt - ½at²
• v² = u² + 2as

What makes these equations so powerful is that each one contains four out of the five motion variables. With knowledge of any three variables, the remaining two can be solved for.

## When to Utilize SUVAT Equations

The SUVAT equations are applicable for motion in a straight line with constant acceleration, where the speed and direction of the body do not change.

## Understanding the Derivation of the Constant Acceleration Equations

Taking a closer look at the process of deriving these equations can provide a deeper understanding of their significance.

### Equation 1: v = u + at

By definition, acceleration is the change in velocity per unit time. This can be expressed as (change in velocity) / (change in time), resulting in the first SUVAT equation: v = u + at. This equation represents the final velocity of a body after accelerating at a constant rate for a certain amount of time.

### Equation 2: s = ½(u + v)t

The average velocity of a body can be determined by taking the average of its initial and final velocities, which is then multiplied by time to obtain the displacement. Therefore, the second SUVAT equation is: s = ½(u + v)t. This equation represents the distance traveled by a body with constant acceleration in a given amount of time.

### Equation 3: s = ut + ½at²

By substituting the value of v from the first equation into the second equation, we can obtain the third SUVAT equation: s = ut + ½at². This equation represents the displacement of a body when its initial velocity, acceleration, and time are known.

### Equation 4: s = vt - ½at²

In a similar manner, by rearranging the first equation and expressing it in terms of u, we get: u = v - at. Substituting this value of u into the third equation results in: s = vt - ½at². This equation represents the displacement of a body when its final velocity, acceleration, and time are known.

### Equation 5: v² = u² + 2as

To obtain the fifth SUVAT equation, we can rearrange the first equation and express it in terms of t: t = (v - u)/a. Substituting this value of t into the second equation, we get: s = ½(u + v)(v - u)/a. Simplifying this equation results in: v² = u² + 2as. This equation represents the final velocity of a body when its initial velocity, acceleration, and displacement are known.

## Solving Motion Problems Using SUVAT Equations

With a sound understanding of how the SUVAT equations are derived, let us now look at some examples of problems that can be solved using these equations.

### Example 1

A car with an initial velocity of 8 m/s is accelerating at a rate of 2 m/s². How long will it take to reach a speed of 20 m/s?

Solution:

Given: v = 20 m/s, u = 8 m/s, a = 2 m/s².

Using the first SUVAT equation: v = u + at.

20 = 8 + 2t

Solving for t, we get t = 6 seconds.

## Understanding Constant Acceleration Equations

Equations are essential tools in determining the movement of objects, as they allow us to calculate variables such as displacement, velocity, acceleration, and time. For situations involving constant acceleration, there are five commonly used equations that can be derived from the fundamental definition of acceleration. Let's take a closer look at these equations and their applications.

## The Equation for Constant Acceleration

The equation v = u + at connects final velocity (v), initial velocity (u), acceleration (a), and time taken (t). It is the fundamental equation for constant acceleration.

## Motion with Constant Acceleration

Aside from the basic equation, there are four other commonly used equations for motion with constant acceleration:

• s = ½ (u + v) t
• s = ut + ½at²
• s = vt - ½at²
• v² = u² + 2as

These equations can be utilized to solve for displacement (s) given any three of the other variables.

## What is Constant Acceleration?

Constant acceleration refers to the change in an object's velocity at a constant rate over time. It is represented by the letter a in the equations and is measured in meters per second squared (m/s²). A classic example of constant acceleration is the free-fall of an object under the influence of gravity, with no other external forces present.

## Examples of Constant Acceleration

Other instances where constant acceleration can be observed include a car accelerating uniformly and a ball rolling down a frictionless ramp. However, it is crucial to note that achieving perfect constant acceleration in reality is challenging since there are usually multiple forces acting on an object.

## The Five Equations of Motion

To summarize, the five equations for motion with constant acceleration are:

• v = u + at
• s = ½ (u + v) t
• s = ut + ½at²
• s = vt - ½at²
• v² = u² + 2as

These equations are valuable in determining various variables in situations involving constant acceleration. Familiarizing oneself with these equations can aid in the study of an object's motion and accurate calculation of its movements.