Linear Motion

In everyday life, we typically think of motion as a movement from one place to another. But to physicists, it is not that simple. Although motion is a movement from one point to another, what type of motion and its plane play an important part in physics.

Motion can be one dimensional, two dimensional, or three dimensional. For this explanation, we look at motion in one dimension, namelymotion (or movement) in a straight line.

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 adisplacement.

Figure 1. Displacement can be in either direction depending on the positive or negative sign. Usama Adeel – StudySmarter Originals

Because displacement is avector 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 Δxis thedisplacement, x_fis the final position, and x_iis the initial position.

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.

The above equation is specifically foraverage velocity, which means it is the calculation of velocity over thewhole 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 thevelocity is the same throughout the displacement, then theaverage velocity equals the instantaneous velocityat any point in time.

Figure 2. Instantaneous velocity will be the same for the duration of displacement if the velocity is constant. Usama Adeel – StudySmarter Originals

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

The gradient of a displacement-time graph

Thegradientat any point in time of adisplacement-time graph is the velocityat that instant.

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

Figure 3. The gradient of a displacement-time graph is velocity. Usama Adeel – StudySmarter Original

To calculate the instantaneous velocity at point p₁,我们take the gradient of the displacement-time curve and make it infinitely small so that it approaches 0. Here’s the calculation, wherex₂is the final displacement, x₁is the initial displacement, t₂is the time at final displacement, and t₁is the time at initial displacement.

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

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:

Just like average velocity, the above equation is foraverage 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

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

If thevelocity of a moving body is the same throughout the displacement, then theinstantaneous acceleration equals zeroat any point in time.

The gradient of a velocity-time graph

Figure 4. The gradient of a velocity-time graph is acceleration. Usama Adeel – StudySmarter Original

In the above velocity-time graph (velocity is on the y-axis and time is on the x-axis), thecurve is the velocity. Let’s say you want to calculate the acceleration at point p₁. The gradient at point p₁is the instantaneous acceleration, and you can calculate it as follows, where v₂is thefinal velocity, v₁is the initial velocity, t₂is thetime at final velocity, and t₁is thetime at initial velocity.

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.

Thefirst equation of motionis

Thesecond equation of motionis

And finally, thethird equation of motionis

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.

Figure 5. Uniform acceleration-time graph. Usama Adeel – StudySmarter Original

Figure 6. Uniform deceleration-time graph. Usama Adeel – StudySmarter Original

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 equationsare enough.

Distance = speed ⋅ time

Displacement = velocity ⋅ time

Linear motion examples