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Velocity and Acceleration

Q: What is velocity and acceleration?

Velocity is defined as the speed with direction and acceleration is the rate of change of velocity with respect to time.

Q: What is the formula for calculating velocity and acceleration?

Velocity is given by change in displacement/time and a cceleration is given by the change in velocity /time.

Q: What is the relationship between velocity and acceleration?

Acceleration is directly proportional to the change in velocity.

Q: What is the difference between acceleration and velocity?

The difference is that they're both measuring different things. 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.

Q: How to calculate average velocity with acceleration and time?

Average velocity is the product of acceleration and the time it takes to achieve that 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!

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Last Updated: 19.06.2022
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Velocity and acceleration formula

The velocity of an object is its speed in a given direction. Velocity is a vector quantity.

Velocity and Acceleration Diagram indicating the difference between speed and velocity StudySmarter

Speed vs velocity, Nidhish Gokuldas StudySmarter originals

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

$V = \frac{∆ s}{t}$

Or in words

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

Where $Δ s$ 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 $m / s$ .

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, $25 m / s$ is a speed, but $25 m / s$ south is a velocity with South being the direction in which the object under consideration is traveling.

Acceleration is the rate of change of velocity with respect to time.

Velocity and Acceleration Diagram showing Acceleration and deceleration StudySmarter 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" $acceleration = \frac{change in velocity}{time taken}$ .

Acceleration is measured in $m / s^{2}$ . 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.

Velocity	Acceleration
Velocity is measured in $m / s$	Acceleration is measured in $m / s^{2}$
A velocity of means that the object is covering a distance of one meter every second.	An acceleration of $1 m / s^{2}$ means that the speed of the object is increasing by $1 m / s$ every second.

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.

$a = \frac{∆ v}{t} = \frac{v - u}{t}$

If $v = u,$ then the acceleration of the body is

$a = \frac{0 m / s}{t} = 0 m / s^{2}$

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 and Acceleration Velocity-Time graph StudySmarter 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.

$gradient = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

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:

$gradient = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{v - u}{t} = \frac{∆ v}{t}$

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

Find the acceleration of the object from the above velocity-time graph for the initial 10 seconds.

The acceleration between two points = gradient of the velocity-time graph

The formula for slope for the initial 10 seconds is given by

$\begin{array}{rcl} a = gradient & = & \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ = & \frac{(5 - 0) m / s}{(10 - 0) s} \\ = & 0.5 m / s^{2} \end{array}$

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.

Velocity and Acceleration Acceleration-time graph StudySmarter

Acceleration-time graph, StudySmarter originals

A car accelerates in $10 s$ from $10 m / s$ to $15 m / s$ . What is the acceleration of the car?

$v = 15 m / s, u = 10 m / s, t = 10 s a = \frac{∆ v}{t} = \frac{v - u}{t} ∆ v = 15 m / s - 10 m / s = 5 m / s a = \frac{15 m / s - 10 m / s}{10 s} a = \frac{5}{10} = 0.5 m / s$

Velocity and Acceleration - Key takeaways

The velocity of an object is its speed in a given direction. Velocity is a vector quantity.
Velocity is given by $V = \frac{∆ s}{t}$ and measured in $m / s$
It is important to mention the direction of motion when we speak of velocity as it is the main differentiating factor between it and speed.
Velocity has a direction as well as magnitude.
Acceleration is the rate of change of velocity with respect to time.
Acceleration is given by $a = \frac{∆ v}{t}$ and is measured in $m / s^{2}$
The Velocity and acceleration of a moving object can be visualized using a time graph.
To calculate the acceleration at any point we need to find the slope of the velocity curve using $a = g r a d i e n t = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} .$
To calculate the velocity from the acceleration-time graph we calculate the area under the acceleration curve.

Flashcards in Velocity and Acceleration

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What is the vertical axis in a velocity-time graph?

velocity

What does a straight line parallel to the x-axis correspond to in a speed-time graph?

No acceleration

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Frequently Asked Questions about Velocity and Acceleration

What is velocity and acceleration?

Velocity is defined as the speed with direction and acceleration is the rate of change of velocity with respect to time.

What is the formula for calculating velocity and acceleration?

Velocity is given by change in displacement/time and acceleration is given by the change in velocity/time.

What is the relationship between velocity and acceleration?

Acceleration is directly proportional to the change in velocity.

What is the difference between acceleration and velocity?

The difference is that they're both measuring different things. 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.

How to calculate average velocity with acceleration and time?

Average velocity is the product of acceleration and the time it takes to achieve that acceleration.

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