(i) Displacement : Displacement is a vector quantity that refers to the shortest distance between the two positions of the object i.e, the difference between the final and initial positions of the object, in a given time. Its d irection is from initial to final position of the object. It is represented by the vector drawn from the initial position to its final position.
(ii) Velocity : Velocity is the vector quantity that refers to “the rate at which an object changes its position.” Imagine a person moving rapidly one step forward and one step back-always returning to the original starting position. While this might result in a frenzy of activity. It would result in a zero velocity. Because the person always returns to the original position.
The time rate of displacement is called velocity
velocity = displacement time
(iii) Acceleration: Generally, the velocity of a moving object changes with time. Sometimes the magnitude of velocity increases and sometimes it decreases.
Sometimes the magnitude remains constant but the direction changes as in circular motion. The rate of change in velocity is defined as acceleration.
Therefore, “The rate of change of velocity of an object with respect to time is known as acceleration.”
In terms of formula:
The unit of acceleration in M.K.S. system is metre/second2(m/s2) Its dimensional formula is [M0L1T-2].
Acceleration is a vector quantity. Similar to velocity it is also divided as follows:
(iv) Speed: Speed of something is the rate at which it moves or travels. Speed is defined as the rate of movement of a body expressed either as the distance travelled divided by the time taken or the rate of change of position with respect to time at a particular point. It is a scalar quantity that refers to “how fast an object is the moving.” “The time rate of distance is called speed.
Speed = \(\frac{\text { distance }}{\text { time }}\)
The unit of speed is m/s. The dimensional formula of speed is [M0LT-1]
(v) Average velocity : Average velocity of a body is defined as the change in position or displacement (Δx) divided by time interval (Δt) in which that displacement occurs.
\(\overrightarrow{v_{a v}}=\frac{\Delta \vec{x}}{\Delta t}=\frac{\overrightarrow{x_{2}}-\overrightarrow{x_{1}}}{t_{2}-t_{1}}\)
(vi) Instantaneous velocity: The instantaneous velocity of a body is the velocity of the body at any instant of time or at any point of its path.
\(\vec{v}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{x}}{\Delta t}=\frac{d \vec{x}}{d t}\)
\(\vec{v}=\frac{d \vec{x}}{d t}\)
Velocity can be positive, negative or zero.
By studying speed and velocity we come to the result that at any time interval average speed of an object is equal or more than the average velocity but instantaneous speed is equal to instantaneous velocity.
(vii) Average Acceleration: “The ratio of total change in velocity to the total time taken is called average acceleration”.
If Δv is the change in velocity in Δt time interval, then;
Average acceleration
(viii) Instantaneous Acceleration : Instantaneous acceleration is defined as “acceleration at any given point or at any instant of time.” If at Δt time interval velocity is Δv then according to above definition; to calculate instantaneous acceleration Δt → 0. Hence,
Instantaneous Acceleration \(a=\lim _{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t}=\frac{d v}{d t}\)
Here, \(\frac{d v}{d t}\), differentiation of v with respect to time dt t which can be known mathematically.
∵ v = \(\frac{dx}{dt}\)
Therefore, \(a=\frac{d}{d t}\left[\frac{d x}{d t}\right]=\frac{d^{2} x}{d t^{2}}\)
Here, \(\frac{d^{2} x}{d t^{2}}\), double differentiation of x w.r.t. t
which can be calculated mathematically.
Therefore, instantaneous acceleration is differentiation of velocity with respect to time and is double differentiation of displacement w.r.t. time.
For any moving object at definite time intervals, if the change in velocity is also same then this is known as uniform acceleration. And if the changes are different in velocity then this is non-uniform acceleration. In same accelerated motion average acceleration and instantaneous acceleration are same.
If in any circular motion (path), the magnitude of velocity of the moving object does not change but the direction of the moving object changes continuously, this type of motion is also called accelerated motion.
Acceleration can be positive, negative or zero. If acceleration is positive its velocity increases. If acceleration is zero then the object moves with a constant speed (velocity). And if the acceleration is negative then the velocity of the object decreases. Hence, negative acceleration is called retardation.
