Kinematics Notes Class 11

Illustration 3 :
A particle is moving in the x-y plane in a circle whose radius is 7 m and centre at the origin. At a particular moment of time coordinates of the particle are (0,7m) and after 5s its coordinates are (7m,0). Find the distance moved by the particle and its displacement in vector form in this 5s.
Solution: Suppose initially the particle is at A and finally it is at B. Two values of distance are possible. If it moves in clockwise sense

The velocity of a particle is the rate of change of its position vector with time whereas speed is the rate of change of distance with time, velocity and speed are two different physical quantities. Former is a vector quantity and the later is a scalar quantity.

(a) Average velocity: Average velocity of a particle is defined over a time interval. Mathematically it is equal to the change in position vector divided by the time interval. In the figure, a particle moving in the x-y plane is at point A(x, y) with position vector at time t = t1, at time t = t 2. The average velocity of the particle over this interval of time is

(b) Instantaneous velocity: Velocity of a particle at a particular instant is known as its instantaneous velocity. In the fig. 4, time taken by the particle to reach the point B from A is

As much smaller the value of that much point B is closer to the point A. Hence velocity of the particle at point A is—

Instantaneous velocity of a particle is the rate of change of its position vector with time.

(c) Average speed: Average speed of a particle over a certain time interval is equal to the distance moved by the particle in that time interval divided by the time interval.

(d) Instantaneous speed: Instantaneous speed of a particle is equal to the magnitude of its instantaneous velocity i.e.,

If distance moved by a particle is known as a function of time by differentiating the function with respect to time we can get the instantaneous speed of the particle.
(Vii) Acceleration: If the velocity of a particle changes continuously either in magnitude or in direction or both then its motion is known as accelerated motion.
(a) Average Acceleration: Average acceleration of a particle over a certain interval of time is equal to the change in its velocity divided by the time interval

(b) Instantaneous Acceleration: The acceleration of a particle at a particular instant is equal to the rate of change of its velocity with time

If the rate of change of velocity of a particle is a constant, then its motion is known as uniformly accelerated motion, in other words, its acceleration is constant.
Acceleration is a vector quantity and a vector quantity is said to be constant if its magnitude and direction both remain same forever.
Note: If the acceleration of a particle is a constant then average acceleration and instantaneous acceleration both are same. Same is true for other physical quantities.

Illustration 4.
A particle in x-y plane moves along a semi circular path of radius R from point A to point B in time t with constant speed v. For the particle calculate
(i) distance travelled
(ii) displacement
(iii) average speed
(iv) average velocity
(v) velocity at A and B
(vi) the speed at A and B
(vii) average acceleration

Solution : (i) Distance = length of semi-circular arc
AB = PR

(ii) Displacement = minimum distance between initial and final point
AB = 2R along y–axis

Illustration 5

The cyclist moves 12 km due north and then due last in 3hr. Find (a) his average speed (b) average velocity, in m/s.
Solution: In the figure, A shows the initial and c the final position of the cyclist. The total distance covered by the cyclist