Velocity and Acceleration



We shall look at the definitions - that is, the meanings - of displacement, speed, velocity and acceleration here.

Displacement

Suppose you move a book on the table 10 cm to your right. We say that the distance moved is 10 cm. However, we would say that the displacement of the book is: 10 cm to your right. In other words, displacement contains two pieces of information: distance and direction.

Suppose I walk 10 m to the north. My distance is just 10 m, but my displacement is: 10 m to the north. You can give the direction in different ways, as long as it is clear.

If a body moves along a straight line, then there are only two directions. One way to indicate the direction is to call one direction plus, and the other minus. For example, if a body moves 2 cm to the right, we may call +2 cm the displacement +2 cm. If it moves 3 cm to the left, we may write down the displacement as -3 cm.

Definition: The displacement of a body is the distance it moves, and the direction that it moves in.

What if I walk 10 m to the right, and then 3 m to the left? What is my total displacement from my start position? The answer is the direct distance from my start position, which is 7 m, and the direction, which is to the right. So this is quite different from the total distance I actually walked, which is 13 m.


Velocity

If you walk 2 m in 1 s, then your speed is 2 m per second. The units for speed are m/s, or m s-1. If a car moves 10 km in one hour, than its speed is 10 km per hour, or 10 km/h.

Definition: The speed of a body is the distance that it moves per unit time.

Unit time just means any convenient length of time - a second, an hour, even a year. For example, the earth makes one turn round the sun per year.

Another way of saying the same is:

Definition: The speed of a body is the rate of change of the distance that it moves.

Formula: v = d/t, where v is the speed, d the distance and t time.

In everyday life, velocity is often used in the same way as speed. In physics, velocity is used to mean the speed and direction of a body. The relation between velocity and speed is like the relation between displacement and distance. Both velocity and displacement contains the additional information of the direction.

Definition: The velocity of a body is the speed that it moves with, and the direction that it moves in.

There is something about velocity that is quite different from displacement. This is the idea of instantaneous velocity and and average velocity.

Imagine a moving car moving at 10 m/s to the right. At a particular instant in time, there is in a zero length of time, the displacement would be zero. However, we can still think of the car as having a velocity of 10 m/s at this instant in time. It does not make sense to try and find the speed by dividing zero distance by zero time, but we can imagine dividing a very short distance like 0.00001 m, by a very short time like 0.000001 s. The answer would be 10 m/s. No matter how short we make this distance and time, we would still get the same answer. So in this sense, it is ok to think that the car has an instantaneous velocity of 10 m/s to the right.

Average velocity, on the other hand, is about the velocity over a certain length of time that is not zero. Suppose that a car moves 100 m to the right in 10 s. It is fast at first, then slows down, then speed up again. The average speed is just the total distance travelled divided by the time taken, that is 100 / 10 = 10 m/s. So the average velocity is 10 m/s to the right. The instantaneous velocity, however, would of course keep changing during the 10 s.

What if the car changes direction? Suppose that in 10 s, the car moves 100 m to the right, and then 10 m to the left. What is the average velocity and speed? The average speed is just total distance over time, that is 110 / 10 = 11 m/s. The average velocity, however, is the total displacement over time. In this case, the total displacement from the start position is 100 - 10 = 90 m. So the average velocity is 90 / 10 = 9 m/s to the right.


Acceleration

If you run faster and faster, you are accelerating. Suppose that you are running at 3 m/s, and that after 1 s you accelerate to 5 m/s. We are talking about instantaneous velocity here of course. We say that your acceleration is 2 m/s per second. That is to say, your velocity increases by 2 m/s in one second. The unit for acceleration is m/s2, or m s-2.

Definition: The acceleration of a body is the change in velocity per unit time.

Formula: a = (v - u)/t, where u is the initial velocity, v is the final velocity, and t the time.

Another way of saying the same is:

Definition: The acceleration of a body is the rate of change of its velocity.




Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 21 July 2011.