Question

Show that if the acceleration is always perpendicular to the speed, the speed is constant.

Answer 1

We have according to the Newtos's 2nd law of motion,

Force acting on a body,

Also,indicates that the direction of acceleration will be in the same direction of the force applied.

Also,we have,velocity of the body,

Since the velocity is the rate of change of displacement,The velocity of the body should have the same direction as the direction of the displacement.

We have the work done by a force causes a displacement is given by,

Where is the angle between the force and displacement.

So,Here given that the acceleration is always perpendicular to the velocity.

So,the angle between the force applied and the displacement should be perpendiculat or

(Since the acceleration and force having the same direction.Also the velocity and displacement has the same direction.)

So,

So,Work done on the body,

So,Since there is no work done by the force on the body.

By work energy theorum,

We have work done on the body by a force is equal to the change in kinetic energies of the body.

Ie,

Let initial velocity, and final velocity,

Here initial kinetic energy,

Final kinetic energy,

ie,

ie, or, ie,final and initial velocities are same.

Here since, , Change in kinetic energy,

So,there is no change in magnitude of velocity or speed.

So,there is no change in kinetic energy of the body or consequently there is no change in magnitude of velocity or the speed of the body.

ie,the speed or the magnitude of the velocity remains constant.