Linear Motion

Linear motion: displacement, velocity, and acceleration

Let’s look at displacement, velocity, and acceleration in more detail.

Displacement

An object can only move in two directions in a straight line, namely forwards or backwards in our case. If we change the position of an object in a particular direction, we are causing a displacement.

Because displacement is a vector quantity, meaning it has a magnitude and a direction, it can be positive or negative. You can take any reference direction as positive or negative, but keep in mind which direction you choose as positive or negative. To calculate displacement, we use the following equation, where Δx is the displacement, x_f is the final position, and x_i is the initial position.

\[\Delta x = \Delta x_f - \Delta x_i\]

Velocity

We can calculate velocity using the following equation, where v is the velocity, Δx is the change in position, and Δt is the change in time.

\[v = \frac{\Delta x}{\Delta t}\]

The above equation is specifically for average velocity, which means it is the calculation of velocity over the whole displacement divided by the total time. But what if you wanted to know the velocity at a certain instant of time and not over the whole period? This is where the concept of instantaneous velocity comes into play.

Instantaneous velocity

We can calculate the instantaneous velocity by applying the average velocity, but we have to narrow the time so that it approaches zero for that particular instant. Now, if you’re thinking that in order to calculate this, you would need to know some calculus, you are right! However, let’s discuss a few scenarios first.

If the velocity is the same throughout the displacement, then the average velocity equals the instantaneous velocity at any point in time.

So, the instantaneous velocity for the above example is 7 m/s (metres per second) as it is not changing at any instant of time.

The gradient of a displacement-time graph

The gradient at any point in time of a displacement-time graph is the velocity at that instant.

Look at the displacement-time graph below with displacement on the y-axis and time on the x-axis. The curve on the graph depicts the displacement over time.

To calculate the instantaneous velocity at point p₁, we take the gradient of the displacement-time curve and make it infinitely small so that it approaches 0. Here’s the calculation, where x₂ is the final displacement, x₁ is the initial displacement, t₂ is the time at final displacement, and t₁ is the time at initial displacement.

Instantaneous velocity at point p₁ \(= \lim_{x \to 0} \frac{\Delta x}{\Delta t} = \frac{x_2-x_1}{t_2-t_1}\)

If the acceleration is constant, we can use one of the kinematics equations (equations of motion) to find the instantaneous velocity. Have a look at the equation below.

\[v = u +at\]

In the above equation, u is the initial velocity, and v is the instantaneous velocity at any instant of time t provided the acceleration remains constant for the whole duration of motion.

Acceleration

We can calculate the acceleration as follows:

\[a = \frac{\Delta v}{\Delta t}\]

Just like average velocity, the above equation is for average acceleration. So what if you wanted to calculate the acceleration at any point in time and not across a period? Let’s look at instantaneous acceleration.

Instantaneous acceleration

A change in velocity at any point in time is instantaneous acceleration. The calculation for instantaneous acceleration is similar to instantaneous velocity.

If the velocity of a moving body is the same throughout the displacement, then the instantaneous acceleration equals zero at any point in time.

The gradient of a velocity-time graph

In the above velocity-time graph (velocity is on the y-axis and time is on the x-axis), the curve is the velocity. Let’s say you want to calculate the acceleration at point p₁. The gradient at point p₁ is the instantaneous acceleration, and you can calculate it as follows, where v₂ is the final velocity, v₁ is the initial velocity, t₂ is the time at final velocity, and t₁ is the time at initial velocity.

Instantaneous acceleration at point p₁ \(= \lim_{v \to 0} \frac{\Delta v}{\Delta t} = \frac{v_2-v_1}{t_2-t_1}\)

Linear motion equations: what are the equations of motion?

The equations of motion govern the motion of an object in one, two, or three dimensions. If you ever want to calculate the position, velocity, acceleration, or even time, then these equations are the way to go.

The first equation of motion is

The second equation of motion is

\[s = ut + \frac{1}{2} at^2\]

And finally, the third equation of motion is

\[v^2 = u^2 + 2as\]

In these equations, v is the final velocity, u is the initial velocity, a is the acceleration, t is time, and s is the displacement.

Important! You can’t use these equations for all motions! The above three equations only work for objects with a uniform acceleration or deceleration.

The graphs below define an object’s uniform acceleration and uniform deceleration.

Also, note that for objects moving with a constant speed and velocity, you don’t need to use the above equations – simple speed and displacement equations are enough.

Distance = speed ⋅ time

Displacement = velocity ⋅ time

Linear motion examples