Velocity and Acceleration

What is velocity? How is it different from speed? What is the point of acceleration if we already know the velocity at which an object is traveling? These are questions you'll be able to answer at the end of this article. First, we go through the definitions and the equations for velocity and acceleration. Then we look at their graphs with respect to time and finally work on a few examples to better our understanding of these concepts. Happy learning!

Velocity and acceleration formula

The velocity of an object is given by the following equation,

$\mathrm{V}=\frac{∆\mathrm{s}}{\mathrm{t}}$

Or in words

$\mathrm{Velocity}=\frac{\mathrm{change}\mathrm{in}\mathrm{displacement}}{\mathrm{time}\mathrm{taken}}$

Where$\Delta s$is the change in displacement (distance in a particular direction) of the object and$t$is the time taken to travel that displacement. Velocity is measured in$\mathrm{m}/\mathrm{s}$.

The main difference between speed and velocity is that speed is a scalar quantity while velocity is a vector quantity. This means that speed only has a magnitude while velocity has a direction as well as a magnitude. For example, $25\mathrm{m}/\mathrm{s}$ is a speed, but$25\mathrm{m}/\mathrm{s}$south is a velocity with South being the direction in which the object under consideration is traveling.

Acceleration and deceleration; when a driver places their foot on the gas pedal, the car will accelerate due to the force the engine exerts on the wheels of the car Nidhish Gokuldas StudySmarter Originals

Given that an object is moving in a straight line with a constant acceleration, we can calculate the object's acceleration if we know how much its velocity changes over a period of time. It is given by the following equation

$a=\frac{∆v}{t}$,

or in words

id="2821967" role="math" $\mathrm{acceleration}=\frac{\mathrm{change}\mathrm{in}\mathrm{velocity}}{\mathrm{time}\mathrm{taken}}$.

Acceleration is measured in$\mathrm{m}/{\mathrm{s}}^2$. Acceleration can be negative or positive. Negative acceleration is called deceleration. Acceleration and velocity are vectors as they have a magnitude and a direction.

Velocity and acceleration difference

Velocity is measuring the rate of change of displacement of a moving object. Acceleration is measuring the rate of change of the velocity of the moving object. The differences between velocity and acceleration are listed below to make the distinction between the two clearer.

Let us now look at the relationship between velocity and acceleration when an object moves in a straight line.

Velocity and acceleration relationship

Looking at the equation for acceleration we can see that it is directly proportional to the change of velocity and inversely proportional to the time it takes to accelerate or decelerate.

• If the velocity of an object is increasing (initial velocity < final velocity) then it has a positive acceleration in the direction of velocity.

• If the velocity is decreasing then the acceleration is negative and in the opposite direction of velocity.

• If the velocity is uniform then the acceleration is zero. This is because acceleration is given by the rate of change in velocity. If the velocity does not change, then the acceleration must be equal to zero.

$\mathrm{a}=\frac{∆\mathrm{v}}{\mathrm{t}}=\frac{\mathrm{v}-\mathrm{u}}{\mathrm{t}}$

If$\mathrm{v}=\mathrm{u},$then the acceleration of the body is

Lets now visualise the relationship between velocity and acceleration of moving objects using graphs.

Velocity and acceleration time graphs

The velocity and acceleration of a moving object can be visualized using a velocity-time graph. The graph below shows the velocity-time graph of an object moving in a straight line.

Velocity-time graph for an object that begins accelerating, then moves at a constant speed before finally decelerating, StudySmarter originals

• The upwards slope of the orange line indicates that the velocity is increasing with respect to time. This means that the object has a positive acceleration.
• The green line is horizontal meaning that the velocity is constant, and therefore the acceleration is zero.
• The blue line is a downward slope that shows the velocity decreasing which is indicative of negative deceleration. To calculate the acceleration at any point we need to find the slope of the velocity curve.

$\mathrm{gradient}\mathit{=}\mathit{}\frac{y_{\mathit{2}}\mathit{-}{y}_{\mathit{1}}}{x_{\mathit{2}}\mathit{-}{x}_{\mathit{1}}}$

where$(x_1,x_2)$are the coordinates of the initial point on the graph and$(x_2,y_2)$are the coordinates of the final point. we know that the Y-axis records velocity and the x-axis records the time taken. So the formula for the gradient of the graph is:

$\mathrm{gradient}\mathit{}\mathit{=}\mathit{}\frac{y_{\mathit{2}}\mathit{-}{y}_{\mathit{1}}}{x_{\mathit{2}}\mathit{-}{x}_{\mathit{1}}}\mathit{=}\mathit{}\frac{v\mathit{-}u}{t}\mathit{=}\frac{\mathit{∆}v}{t}$

Let us look at an example of how to read an acceleration-time graph.

Now let us plot the acceleration time graph for the same object in the timeframe of$0‐30$seconds. The acceleration is recorded on the y-axis, and the time is recorded on the x-axis.