Average Acceleration and Accelerated Motion

Average Acceleration: If the velocity of an object changes, it is said to be. Moving with an acceleration.

The time rate of change of velocity of a body is called acceleration.

As velocity is a vector so an change in velocity may be due to change in its magnitude or a change in its direction or both.

Consider a body whose velocity v1 at any instant t changes to v2 in further small time interval ∆t. the two velocity vectors v1 and v2 and the change in velocity, v2 – v1 = ∆ v, are represented in fig. 3.3. the average acceleration  during time interval ∆t is given by

As  is the difference of two vectors divided by a scalar ∆t,  must also be a vector. Its direction is the same as that of ∆v. acceleration of a body at a particular instant is known as instantaneous acceleration and it is the value obtained from the average acceleration as ∆t is made smaller and smaller till it approaches zero. Mathematically. It is expressed as

Instantaneous acceleration

If the velocity of a body is increasing, its acceleration is positive but if the velocity is decreasing the acceleration is negative. If the velocity of the body changes by equal amount in equal intervals of time, the body is said to have uniform acceleration. For a body moving with uniform acceleration, its average acceleration is equal to instantaneous acceleration.

Review of equations of uniformly accelerated motion

In school physics we have studied some useful equations for objects moving at constant acceleration.

Suppose an object is moving with uniform acceleration a along a straight line. If is initial velocity is v, and final velocity after a time interval t is v1. Let the distance covered during this interval be S the we have

These equations are useful only for linear motion with uniform acceleration. When the object moves along a straight line, the direction of motion does not change. In this case all the vectors can be manipulated like scalars. In such problems, the direction of initial velocity is taken as positive. A negative sign is assigned to quantities where direction is opposite to that of initial velocity.

The equation for uniformly accelerated motion can also be applied to free fall motion of the objects by replacing a by g.

Do You know?

How the displacement of a vertically thrown ball varies with time?

How the velocity of a vertically thrown ball varies with time? Velocity is upwards positive.

Do You Know?

At the surface of the earth, in situations where air friction is negligible, objects fall with the same acceleration regardless of their weights.

Newtons Laws of Motion

Newtons Laws are empirical laws, deduced from experiments. They were clearly stated for the first time by sir Isaac Newton, who published them in 1687 in his famous book called “principia”. Newton’s laws are adequate for speeds that are low compared with the speed of light. For very fast moving objects, such as atomic particles in an accelerator, relativistic mechanics developed by Albert Einstein is applicable.

You have already studied these laws in your secondary school physics. However a summarized review is given below.

 Newton’s First Law of motion

A body at rest will remain at rest, and a body moving with uniform velocity will continue to do so, unless acted upon by some unbalanced external force. This is also known as law of inertia. The property of an object tending to maintain the state of rest or state of uniform motion is referred to as the object’s inertia. The more inertia, the stronger is this tendency in the presence of a force. Thus,

The frame of reference in which Newtons first law of motion holds, is known as inertial frame of reference. A frame of reference stationed on earth is approximately an inertial frame of reference.

Newton’s Second Law of Motion

A force applied on a body produces acceleration in its own direction. The acceleration produced varies directly with the applied force and inversely with the mass of the body. Mathematically, it is expressed as

F = ma                   ………………..         (3.9)

Newton’s Third Law of Motion

Action and reaction are equal and opposite. For example, whenever an interaction occurs between two objects, each object exerts the same force on the other, but in the opposite direction and for the same length of time. Each force in action-reaction pair acts only on one of the two bodies, the action and reaction forces never act on the same body.