## Sunday, February 22, 2015

### Average Speed Average Velocity and Acceleration

Average speed and average velocity

The average speed is defined as the ratio of the total distance traveled by the body in the total given time. The average velocity is defined as the ratio of the total displacement covered by a body in the total time.

Speed is a scalar quantity which has only a magnitude but not specified direction. Velocity is a vector quantity which has both magnitude and direction and also satisfies the laws of the vectors.

In any given situation we can calculate the average speed and average velocity. If a body comes back to its original position after a certain time, its average velocity can become zero but it will have its average speed.

When we are talking about a straight-line motion where the body is going to have only a forward motion, there won’t be any significant difference in terms of the magnitude of average speed and average velocity. In a given situation we have to calculate average speed or average velocity.

Problem and solution

A body moves half of the time with one velocity and during the other half of the time it moves with a different velocity. Find the average of those velocities?

Basing on the definition of average velocity, we can write it as the ratio of total displacement to the total given time. Let in the first half of the time it covers a specific distance and the remaining half of the time it is covering some another distance. The total distance is the sum of these two distances. The total time ease some of the two halves of the time which is equal to the total time of the journey itself. As distance is not given in the problem, we have to express it in terms of velocity and time. We know that the distance is defined as the product of velocity and time and taking that into consideration, we can derive the equation for the average velocity of the above case as shown below.

Problem and solution

If a body covers half of the displacement with one velocity and the second half of the displacement with some another velocity, find its average velocity?

We can solve this problem also basing on the same formula as the average velocity is the ratio of total displacement to the total time. But solving this problem is slightly different because if the distance is given and time is not given.

Here we have to convert the time in terms of displacement and velocity and the problem is solved as shown below.

Thus the average velocity into different situations could be different.

Problem and solution

A body is covering three equal parts of the total displacement with three different velocities. Find its average velocity?

This problem is also solved under the same lines, basing on the very definition of the average velocity is the total displacement per unit time. The solution is as attached below.

Acceleration

At the broader level, we can define the acceleration as the rate of change of velocity. We can define instantaneous acceleration as the rate of change of velocity at a very small interval of time which intends to 0.

If the velocity is varying uniformly with respect to time, it’s acceleration is uniform. If velocity is not varying uniformly with respect to time, it’s acceleration is non uniform and in that case we may need to integrate the equation to get the actual acceleration.
If a body is having a uniform velocity, it means it is not having acceleration. All the bodies of the Earth experience acceleration due to the gravitational force of the Earth and this acceleration is called acceleration due to gravity.

The numerical value of acceleration due to gravity is constant and it is equal to 9.8 m/s Squire. It is always directed towards the Center of the Earth and it is due to the mass as well as the size of the Earth.

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