Understanding Acceleration and Motion

1. Instantaneous Velocity

When an object moves, its velocity at any exact moment is called instantaneous velocity. To find it:

  • Calculate the average velocity over smaller and smaller time intervals (\Delta t).
  • The limiting value of \(\frac{\Delta x}{\Delta t} as \Delta t \to 0\) gives the instantaneous velocity: v = \frac{dx}{dt}.

Example: If position x(t) = 8.5 + 2.5t^2, then velocity v = \frac{dx}{dt} = 5t. At t = 2\ \text{s}, v = 10\ \text{m/s}.

2. Average Velocity vs. Speed

  • Average velocity = \frac{\Delta x}{\Delta t} (direction matters).
  • Speed is just the magnitude of velocity (e.g., both +24\ \text{m/s} and -24\ \text{m/s} have a speed of 24\ \text{m/s}).

3. Acceleration

Acceleration describes how velocity changes over time:

  • Average acceleration: \overline{a} = \frac{\Delta v}{\Delta t}.
  • Instantaneous acceleration: a = \frac{dv}{dt} (slope of the v-t graph).

Acceleration can be positive (speeding up), negative (slowing down), or zero (constant velocity).

4. Motion Graphs

  • Position-time (x-t) graph:
    • Curves upward for positive acceleration.
    • Curves downward for negative acceleration.
    • Straight line for zero acceleration.
  • Velocity-time (v-t) graph:
    • Area under the curve = displacement.
    • Slope = acceleration.

5. Kinematic Equations (Constant Acceleration)

For objects with uniform acceleration:

  • v = v_0 + at (final velocity).
  • x = v_0t + \frac{1}{2}at^2 (displacement).
  • v^2 = v_0^2 + 2ax (velocity-displacement relation).

Key note: Average velocity for constant acceleration is \overline{v} = \frac{v + v_0}{2}.

Important Concepts for NEET

  1. Instantaneous velocity and acceleration: Definitions and calculations using calculus or graphs.
  2. Motion graphs: Interpreting x-t, v-t, and a-t graphs for direction and magnitude changes.
  3. Kinematic equations: Solving problems with constant acceleration (e.g., free-fall, inclined planes).
  4. Speed vs. velocity: Understanding the difference (magnitude vs. magnitude + direction).
  5. Area under v-t curve: Calculating displacement directly from the graph.

Example Problem

An object moves along the x-axis with position x(t) = 8.5 + 2.5t^2. Find:

  1. Velocity at t = 0\ \text{s} and t = 2\ \text{s}.
  2. Average velocity between t = 2\ \text{s} and t = 4\ \text{s}.

Solution:

  1. Velocity: v = \frac{dx}{dt} = 5t → At t = 0\ \text{s}, v = 0\ \text{m/s}; at t = 2\ \text{s}, v = 10\ \text{m/s}.
  2. Average velocity: \overline{v} = \frac{x(4) – x(2)}{4 – 2} = 15\ \text{m/s}.

Remember: Acceleration and velocity can’t change abruptly—they vary smoothly over time!