Speed, Velocity, Acceleration and Retardation: Definitions, Units, Dimensions & Solved Examples

Speed, Velocity, Acceleration and Retardation: Definition, Units, Dimensions & Numerical Problems

This page provides detailed explanations of speed, velocity, acceleration, and retardation, including definitions, SI and CGS units, dimensional formulas, important questions, and step-by-step solved numerical problems for better understanding.

Speed

Definition: The distance travelled by an object in unit time is called its speed.

Formula: \( v = \frac{x}{t} \), where:

• \(v\): speed of the body
• \(x\): distance travelled
• \(t\): time taken

Type: Scalar quantity (has magnitude only)

Instantaneous Speed

The speed of a body at a particular instant of time is called instantaneous speed.

\(v=\frac{dx}{dt}\)

Units of Speed

• CGS Unit: cm/s
• SI Unit: m/s

Dimension: \([M^0 L T^{-1}]\)

Instrument: Speedometer in a car shows instantaneous speed.

Frequently Asked Questions on Speed

1. How is average speed calculated?
   \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)
2. What is uniform speed?
   If a body covers equal distances in equal intervals of time, its speed is uniform.
3. What is instantaneous speed?
   Speed at a specific instant of time.
4. When are average speed and instantaneous speed equal?
   When the body moves with uniform speed.

Velocity

Definition: The displacement of an object per unit time is called its velocity.

If the displacement of a body in time t is s then velocity \(v=\frac{s}{t}\)

instantaneous Velocity \( v = \frac{\Delta s}{\Delta t}=\frac{ds}{dt} \)

Type: Vector quantity (has magnitude and direction)

Units of Velocity

• CGS Unit: cm/s
• SI Unit: m/s

Dimension: \([M^0 L T^{-1}]\)

Difference Between Speed and Velocity

Speed	Velocity
Distance per unit time	Displacement per unit time
Scalar quantity	Vector quantity
Average speed cannot be zero	Average velocity can be zero

Frequently Asked Questions on Velocity

1. What does the slope of a position-time graph represent?
   Velocity
2. What does the area under a velocity-time graph represent?
   Displacement
3. Slope of velocity-time graph?
   Acceleration
4. Area under acceleration-time graph?
   Change in velocity
5. When are instantaneous and average velocity equal?
   For uniform motion

Acceleration

Definition: The rate of change of velocity with respect to time is called acceleration.

Formula: \( a = \frac{dv}{dt} \)

Type: Vector quantity

Units of Acceleration

• CGS Unit: cm/s²
• SI Unit: m/s²

Dimension: \([M^0 L T^{-2}]\)

Retardation (Negative Acceleration)

When the velocity of a body decreases with time, it is called retardation.

Important FAQs on Acceleration and Retardation

1. When instantaneous and average acceleration are equal?
   Ans: For uniformly accelerated motion.
2. Can acceleration be negative?
   Ans: Yes, during retardation.
3. Can velocity be zero but acceleration non-zero?
   Ans: Yes, e.g., at the topmost point in vertical motion.
4. Speed constant but velocity changing — possible?
   Ans: Yes, in uniform circular motion.
5. Constant velocity but variable acceleration — possible?
   Ans: No.
6. Ways velocity can change:
   • Only magnitude changes
   • Only direction changes
   • Both magnitude and direction change
7. Examples:
   • Velocity zero but acceleration ≠ 0: topmost point of upward throw
   • Acceleration opposite to velocity: upward motion under gravity
   • Velocity perpendicular to acceleration: uniform circular motion

Numerical Problems and Examples

Example 1:

Ratan goes to market at 40 km/h and returns at 30 km/h. Find average speed.

Solution:

Let distance = \( x \). Time taken to go = \( \frac{x}{40} \), time to return = \( \frac{x}{30} \)

Total distance = \( 2x \), total time = \( \frac{x}{40} + \frac{x}{30} \)

Average speed:
\[ \text{Avg Speed} = \frac{2x}{\frac{x}{40} + \frac{x}{30}} = \frac{2}{\frac{1}{40} + \frac{1}{30}} = 34.28 \text{ km/h} \]

Example 2:

Position function: \( x = 3t^2 + 4t + 2 \)

1. Instantaneous velocity: \( v = \frac{dx}{dt} = 6t + 4 \)
2. Not uniform (depends on time)
3. At \( t = 2 \): \( v = 6(2) + 4 = 16 \text{ m/s} \)
4. Acceleration: \( a = \frac{dv}{dt} = 6 \text{ m/s}^2 \)

Example 3:

Two bodies A and B make angles 30° and 60° with the time axis. Find ratio of velocities.

Solution:
\[ \text{Ratio of velocities} = \tan(30^\circ) : \tan(60^\circ) = \frac{1}{\sqrt{3}} : \sqrt{3} = 1 : 3 \]

Equations of Uniformly Accelerated Motion

When acceleration is constant:

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)

For retardation, use \( a \rightarrow -a \). For free fall, use \( a = g \).

Example 4:

1. \( s = \frac{1}{2}at^2 \) when \( u = 0 \) ⇒ \( s \propto t^2 \)
2. Distances in 1s, 2s, 3s: \( 1:4:9 \)
3. Distances in 1st, 2nd, 3rd second: \( 1:3:5 \)

Example 5:

If a car stops in distance \( S \) when moving with speed \( V \), show that with speed \( nV \), the stopping distance is \( n^2S \).

Using: \( v^2 = u^2 - 2aS \)

Initial case: \( 0 = V^2 - 2aS \Rightarrow a = \frac{V^2}{2S} \)

For speed \( nV \):
\[ 0 = (nV)^2 - 2aS' \Rightarrow S' = n^2 S \]

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