Revision: Class 11 >> Motion in One Dimension NEET (UG) Motion in One Dimension

"total path length travelled during the time interval over which average speed is being calculated, divided by that time interval."

"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."

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

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.

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

“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.

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.

Relative velocity is the velocity of one object as measured from another moving object's perspective.

Let:

• v_A = velocity of object A (relative to ground/Earth)
• v_B = velocity of object B (relative to ground/Earth)
• v_AB = velocity of A relative to B (what B observes about A's motion)

Average Speed = v_av = \[\frac{\text{path length}}{\text{time interval}}\]

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

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

Dimensions: [L¹M⁰T⁻¹]

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

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

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.

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

v_AB = v_A - v_B

v_BA = v_B - v_A = -v_AB

Key relationship: v_AB = -v_BA