# Kinematics in One Dimension

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This article is a topic within the subject Higher Physics 1A.

## Introduction

Kinematics is the study of motion. In one dimension, objects are treated as though they only travel in a single direction.

## Displacement, velocity and acceleration

 Displacement, velocity and acceleration are key concepts in kinematics:

• A particle's position is measured as displacement from the origin of the coordinate system. Displacement is measured in meters (m).
• Velocity is the rate of change of displacement over time, measured in meters per second (ms-1). Velocity is the derivative of displacement with respect to time.
• Acceleration is the change in velocity over time, measured in meters per second per second (ms-2). Acceleration is the derivative of velocity with respect to time.

All of these are vector quantities. This means they have both magnitude (size) and direction.

### Notation

• s = displacement.
• v = ds/dt = velocity
• a = dv/dt = d2s/dt2 = acceleration

The symbols shown above will often be accompanied by subscripts to indicate their direction. For example, vx0 refers to the velocity in the x direction at time t=0.

Another form of notation is to replace "s" with the direction of displacement. Displacement in the x direction thus becomes "x". Velocity and acceleration are indicated using dots or derivatives. Subscripts are still used to indicate time, where necessary. For example:

• x = displacement (x direction)
• ẋ = dx/dt = velocity (x direction)
• ẍ = d2x/dt2 = acceleration (x direction)

## Motion with Constant Acceleration

### Constant Acceleration Formulas

 vxt = vx0 + axt

Velocity at time t, expressed in terms of initial velocity (vx0) and constant acceleration (axt)

vx,avg = (vx,i + vx,f)/2

Average velocity is the average of initial and final velocity.

xt = x0 + vx0t + ½at2

Displacement at time t, expressed in terms of initial displacement (x0), displacement due to initial velocity (vx0t) and displacement due to constant acceleration (½at2)

xt = x0 + ½(vx0 + vxt)t

Displacement at time t, expressed in terms of initial displacement (x0), initial velocity (vx0), final velocity (vxt) and time (t)

vxt2 = vx02 +2ax(xt-x0)

Velocity at time t, expressed in terms of initial velocity, acceleration, and particle location.

### Using the Formulas

 The best approach in all physics problems is to try and understand the situation first. Once you have done this, you should be able to decide on appropriate formulas to use, and whether you will need calculus, simultaneous equations, etc. Make sure that the formulas you choose make sense for the situation - for instance, the formulas above cannot be applied to a vehicle with changing acceleration. Applying formulas blindly is risky because you will often miss important aspects of the question or apply inappropriate formulas.

## Non-uniform Acceleration

Non-uniform acceleration occurs when the acceleration varies over time. This means that the formulas for constant acceleration can no longer be used. There are several common situations where non-uniform acceleration may occur:

• The action of varying forces will cause varying acceleration (since F=ma). Forces are discussed in Particle Dynamics.
• Circular motion involves changing acceleration. Circular motion may be uniform (only the direction of acceleration changes) or non-uniform (direction and magnitude of acceleration change).
• Acceleration may vary with time or other parameters.

## End

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