Unit 2.4 – Equations of Motion

• 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 ($𝑎$) = $\frac{\mathrm{\Delta }𝑣}{\mathrm{\Delta }𝑡}=\frac{0}{𝑡}=0{\text{m/s}}^2$.