Acceleration

We see objects moving around. These objects do not always move at a constant speed, or in a constant direction. For example, planets revolving around the sun change their direction at every point, cars moving on a road subjected to traffic have different speeds at different point of time. How do we quantize this change in their velocity, both the magnitude and direction? Here, we introduce the term acceleration, which is the rate of change of velocity of an object. But how does it work in terms of change in direction of velocity?

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Whenever we are getting late for school and your father is driving you to school, you ask your father to hurry up and in response to that, you feel the car going faster. This is called acceleration. So what is acceleration?

From the above observation, it can be said as the vehicle’s capacity to gain speed. It is actually a change in velocity. Mathematically, it is defined as a rate of change of velocity.

An equation can be represented as:

Acceleration (a) = \( \frac {change~in~velocity}{Time~Taken}\)

\(~~~~~~~~~~~~~~~~~~~~\)   ⇒ a = \( \frac {v~-~u}{t} \)

So unit for acceleration is m/s2. It can also be interpreted using Newton’s second law i.e.

F = ma

Hence a body is accelerated only if a net external force acts on the body. And the acceleration produced is inversely proportional to mass. Based on the direction of velocity it can be positive or negative. When both act in the same direction then it is said as positive or the velocity of the body increases. Like if I press accelerator pedal of the vehicle the velocity increases. Similarly, when they act in opposite directions the acceleration is said to be negative or the velocity decreases. An example is, braking pedal of a vehicle when we press it the velocity decreases. There is another classification for it that is uniform and non – uniform. If the acceleration remains constant for a given interval of time it is said to be uniform while if it changes it is said to be non – uniform. When it is uniform, we can use equations of motion. We should keep in mind that these equations can’t be used when a body is accelerated non-uniformity. Graphically they can be represented as follows.

Acceleration Uniform and Non-uniform acceleration

So can we have a situation when speed remains constant but the body is accelerated? Actually, it is possible in circular where speed remains constant but since the direction is changing hence the velocity changes and the body is said to be accelerated.

Average acceleration

The average acceleration over a period of time is defined as the total change in velocity in the given interval divided by the total time taken for the change. For a given interval of time, it is denoted as ā.

Mathematically, acceleration

Where v2 and v1 are the instantaneous velocities at time t2 and t1 and ā is the average acceleration.

Velocity-time graph

acceleration

Average acceleration: In the velocity-time graph shown above, the slope of the line between the time interval t1 and t2 gives the average value for the rate of change of velocity for the object during the time t1and t2.

Instantaneous acceleration: In a velocity-time curve, the instantaneous acceleration is given by the slope of the tangent on the v-t curve at any instant.

acceleration

Positive, negative and zero acceleration

acceleration

Consider the velocity-time graph shown above. Here, between the time intervals of 0-2 seconds, the velocity of the particle is increasing with respect to time; hence the body is experiencing a positive acceleration as the slope of the v-t curve in this time interval is positive.

Between the time intervals of 2-3 seconds, the velocity of the object is constant with respect to time; hence the body is experiencing zero acceleration as the slope of the v-t curve in this time interval is 0.

Now, between the time intervals of 3-5 seconds, the velocity of the body is decreasing with respect to time; hence the body experiences a negative value of the rate of change of velocity as the slope of the v-t curve in this time interval is negative.

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Practise This Question

In the following four periods
(i) Time of revolution of a satellite just above the earth’s surface (Tst)
(ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth (Tma)
(iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/k (Tsp)
(iv) Period of an infinite length simple pendulum in the earth’s real gravitational field (Tis)