**Subject :**Elements of Mechanical Engineering

## Kinematics of a Particle

**Kinematics of a Particle:**** **Kinematics is the area of mechanics concerned with the study of motion of particles and rigid bodies without consideration of what has caused the motion. When we take into consideration the factors causing the motion, the area of study is called dynamics.

A particle has no size but has mass. This is a completely hypothetical concept. However, many times object can be approximated as particle. For example, a car is moving. Compared to distance, it travels, its size is very small and it can be treated as particle.

Moreover, a rigid body can be considered as a combination of small particles. Thus, the concepts learned for a particle can be helpful in understanding the kinematics of rigid body.

**Velocity and Acceleration of a particle: **If the path of motion of a particle is known i.e., one knows the positon of the particle as a function of time, the velocity of the particle can be calculated as follows.

Velocity is the rate of change of the position of the particle. In the xyz reference system, if at time *t* the particle is at *P* with position vector r(t) and time , the particle is at *P*' with position vector

r (), then the velocity vector v is,

In the figure is the displacement given by the chord PP' and is the distance given by arc PP'. In the limit, becomes a unit vector tangent to the trajectory.

Thus,

Therefore, velocity of a particle is a vector having a magnitude equal to the speed of the particle and a direction tangent to the trajectory. The acceleration vector of a particle can be given as,

We can obtain a also in the following manner,

Note that, we have indicated the first derivative by putting a dot on the variable. Also,

where double dot indicates the double differentiation with respect to time.

If the acceleration is given as a function of time, velocity and displacement can be found by integration.

Example: A particle moves in a straight line from rest with an acceleration of t, where t is the time. In 5 seconds, where will the particle be from the starting point?

Solution: Here, the motion is one dimensional.