Unit 2.3 – Acceleration

• Definition: Acceleration is the rate of change of velocity with respect to time.
• Nature: It is a vector quantity, meaning it has both magnitude and direction.
• Formula:
  $$\text{Acceleration}=\frac{\text{Change in Velocity}}{\text{Time Taken}}=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$$
  Where:
  • $\mathrm{\Delta }v=v_f-v_i$ (final velocity − initial velocity).
  • $\mathrm{\Delta }t$ = time interval.
• SI Unit: Meter per second squared (${\text{m/s}}^2$).

• Uniform Acceleration: When the velocity changes by equal amounts in equal intervals of time.
  Example: A car accelerating at $2{\text{m/s}}^2$.
• Variable Acceleration: When the velocity changes unequally over time.
  Example: A rocket taking off with increasing thrust.
• Deceleration (Negative Acceleration): When the velocity decreases over time.
  Example: A car braking to stop.

• Acceleration can be positive, negative, or zero.
  • Positive acceleration: Velocity increases.
  • Negative acceleration (deceleration): Velocity decreases.
  • Zero acceleration: Velocity remains constant (uniform motion).
• Acceleration depends on the change in velocity, not speed.

Quick Revision Points

• Acceleration: Rate of change of velocity; vector; ${\text{m/s}}^2$.
• Formula: $a=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{v_f-v_i}{t}$.
• Uniform acceleration: Equal velocity changes in equal time intervals.
• Deceleration: Negative acceleration; velocity decreases.
• Zero acceleration: Constant velocity (no change in velocity).

Previous Year Questions and Answers

• Initial velocity ($v_i$) = $0\text{m/s}$.
• Final velocity ($v_f$) = $20\text{m/s}$.
• Time ($t$) = $5\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{20-0}{5}=4{\text{m/s}}^2$.

• Initial velocity ($v_i$) = $30\text{m/s}$.
• Final velocity ($v_f$) = $0\text{m/s}$.
• Time ($t$) = $10\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{0-30}{10}=-3{\text{m/s}}^2$.
• Deceleration = $3{\text{m/s}}^2$.

• Initial velocity ($v_i$) = $10\text{m/s}$.
• Final velocity ($v_f$) = $25\text{m/s}$.
• Time ($t$) = $5\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{25-10}{5}=3{\text{m/s}}^2$.

• Since the velocity is constant, there is no change in velocity ($\mathrm{\Delta }v=0$).
• Acceleration ($a$) = $\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{0}{t}=0{\text{m/s}}^2$.

• Initial velocity ($v_i$) = $0\text{m/s}$ (dropped from rest).
• Final velocity ($v_f$) = $20\text{m/s}$.
• Time ($t$) = $2\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{20-0}{2}=10{\text{m/s}}^2$.

Expected Questions

• Initial velocity ($v_i$) = $15\text{m/s}$.
• Final velocity ($v_f$) = $35\text{m/s}$.
• Time ($t$) = $4\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{35-15}{4}=5{\text{m/s}}^2$.

• Initial velocity ($v_i$) = $50\text{m/s}$.
• Final velocity ($v_f$) = $20\text{m/s}$.
• Time ($t$) = $6\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{20-50}{6}=-5{\text{m/s}}^2$.
• Deceleration = $5{\text{m/s}}^2$.

• Initial velocity ($v_i$) = $0\text{m/s}$.
• Final velocity ($v_f$) = $12\text{m/s}$.
• Time ($t$) = $3\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{12-0}{3}=4{\text{m/s}}^2$.

• Initial velocity ($v_i$) = $30\text{m/s}$.
• Final velocity ($v_f$) = $0\text{m/s}$.
• Time ($t$) = $3\text{s}$.
• Acceleration ($a$) = $\frac{v_f-v_i}{t}=\frac{0-30}{3}=-10{\text{m/s}}^2$.
• Deceleration = $10{\text{m/s}}^2$.

• Since the velocity is constant, there is no change in velocity ($\mathrm{\Delta }v=0$).
• Acceleration ($a$) = $\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{0}{t}=0{\text{m/s}}^2$.