Revision: Kinematics >> Motion in a Straight Line Physics Science (English Medium) Class 11 CBSE

"Average velocity is defined as the displacement of the object during the time interval over which average velocity is being calculated, divided by that time interval."

OR

The total displacement Δ\[\vec x\] of an object divided by the total time interval Δt over which that displacement occurs is called average velocity.

Instantaneous velocity of an object is its velocity at a given instant of time. It is defined as the limiting value of the average velocity of the object over a small time interval (Δt) around t when the value of the time interval (Δt) goes to zero.

OR

The limiting value of the average velocity of an object over a small time interval 'ΔtΔt' around time tt when the value of the time interval goes to zero is called instantaneous velocity.

Instantaneous speed is simply the speed of an object at a single, specific moment in time (t).

OR

The limiting value of the average speed of an object over a small time interval 'Δt' around time tt when the value of the time interval goes to zero is called instantaneous speed.

Acceleration is defined as the rate of change of velocity with time.

The change in velocity of an object divided by the total time required for that change in velocity is called average acceleration.

The limiting value of the average acceleration of an object over a small time interval 'Δt' around time tt when the value of the time interval goes to zero is called instantaneous acceleration.

Non-uniform motion is used to mean the movement in which the object does not cover the same distance in the same distances in the same time intervals, regardless of the length of the time intervals. Every time the speed of the moving object changes by a different proportion at the same time interval, the motion of the body is observed as non-uniform motion.

For example:

1. A horse running.
2. A bouncy ball.
3. A car coming to a halt.

“In physics, uniform motion is defined as the motion where the velocity of the body travelling in a straight line remains the same. When the distance travelled by a moving thing is the same at several time intervals, regardless of the time length, the motion is said to be uniform motion.”

For example,

• The hour hand of the clock: It moves with uniform speed, completing movement of a specific distance in an hour.
• An aeroplane is cruising at a level height and a steady speed.
• A car is going along a straight, level road at a steady speed.

\[\vec{v}_{\mathrm{av}}=\frac{\vec{x}_2-\vec{x}_1}{t_2-t_1}\]

• v_av : average velocity.
• x₂ : final position vector.
• x₁ : initial position vector.
• t₂ : final time
• t₁ : initial time

Dimensions: [L¹M⁰T⁻¹]

OR

Average Velocity: \[\vec V_{avg}\] = \[\frac {\text {Displacement}}{\text {Time interval}}\] = \[\frac {x_2-x_1}{t_2-t_1}\] = \[\frac {Δ\vec x}{Δt}\]

\[\vec{\mathrm{v}}=\lim_{\Delta t\to0}\left(\frac{\Delta\vec{x}}{\Delta t}\right)=\frac{d\vec{x}}{dt}\]

To calculate instantaneous speed, we look at the average speed (Average Speed=ΔtDistance​) over a very, very short time interval (Δt). It is defined as the limiting value of the average speed as the time interval (Δt) approaches zero.

Instantaneous Speed = \[\operatorname*{lim}_{\Delta t\to0}\frac{\mathrm{Distance}}{\Delta t}\]

OR

\[\vec{\mathbf{v}}=\lim_{\Delta t\to0}\frac{\Delta\vec{\mathbf{x}}}{\Delta t}=\frac{d\vec{\mathbf{x}}}{dt}\]

Average acceleration is calculated when an object has velocities \[\vec v_1\] and \[\vec v_2\] at times t₁ and t₂:

\[\vec{a}=\frac{\vec{v_2}-\vec{v_1}}{t_2-t_1}\]

where:

• \[\vec a\] = average acceleration
• \[\vec v_1\] = velocity at time t₁
• \[\vec v_2\] = velocity at time t₂

OR

Average acceleration: \[\vec a_{av}=\frac {\vec v_2-\vec v_1}{t_{2}-t_{1}}=\frac {\Delta\vec v}{\Delta t}\]

Instantaneous acceleration is the limiting value of average acceleration when the time interval approaches zero:

\[\vec{a}=\lim_{\Delta t\to0}\frac{\Delta\vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]

where:

• \[\vec a\] = instantaneous acceleration
• \[d\vec{v}\] = infinitesimal change in velocity
• dt = infinitesimal change in time

The instantaneous acceleration at a given time equals the slope of the tangent to the velocity versus time curve at that time.

• Slope of x-t graph = Velocity → Horizontal line (x = constant) = rest; positive slope = uniform velocity along +X axis; negative slope = motion along −X axis; curve = non-uniform velocity.
• Slope of v-t graph = Acceleration → Horizontal line = zero acceleration (constant velocity); positive slope = constant positive acceleration; negative slope = constant negative acceleration; curve = non-uniform acceleration.
• Area under v-t graph = Displacement of the object during that time interval.
• x-t graph shapes → Zigzag/oscillatory curve = oscillatory motion with constant speed; ever-increasing curve = accelerated (non-uniform) motion.
• Relative Velocity → \[\vec v_{AB}\] = \[\vec v_A\] − \[\vec v_B\]​ and \[\vec v_{BA}\] = \[\vec v_B\] − \[\vec v_A\]; they are always equal in magnitude and opposite in direction.