Position, Velocity and Acceleration in Curvilinear Motion

Position, Velocity and Acceleration in Curvilinear Motion

• For a particle moving along a curved path in a plane, the position, velocity and acceleration are vector quantities.

• Consider a particle moving along a curved path as shown in Fig. 9.5.1. If the particle is at point 'P' at time 't', then its position is defined by vector which is directed from O to P.

• Let the particle be at position P' at time t + Δt.

The position vector at P' is   is the change in position i.e. displacement of the particle represented by the vector

• The distance travelled Δs is the length of arc PP'.

will be the average velocity and the instantaneous velocity is

• The magnitude of is called speed.

As Δt → 0, P' is very close to P and the direction of is tangential to the curve.

• Hence velocity of the particle is always tangential.

• Consider velocity vectors at two points A and B as shown in Fig. 9.5.3. They are redrawn from a common point as shown in Fig. 9.5.4.

• The change in velocity is

• The average acceleration is

and instantaneous acceleration is

• The three vectors are shown in Fig. 9.5.5. Note that the O velocity is tangential whereas acceleration is not.