Comprehensive Notes by Mohit Sir (MSc)
An object is said to be in motion when its position changes with time relative to a reference point (frame of reference).
Reference Point: A fixed point or object with respect to which the position of other objects is described (e.g., a tree, building, or stationary object).
Have only magnitude (no direction)
Have both magnitude and direction
Distance moved is the actual length of the path travelled by a body.
Displacement is the length of the shortest path travelled by a body from its initial position to its final position.
| Distance | Displacement |
|---|---|
| Total path length covered | Shortest distance between initial and final positions |
| Scalar quantity | Vector quantity |
| Always positive | Can be positive, negative or zero |
| Depends on path | Independent of path |
Example: If a person walks 3km East and then 4km North:
Distance = 3 + 4 = 7km
Displacement = √(3² + 4²) = 5km Northeast
If a body travels equal distances in equal intervals of time, it is said to be in uniform motion.
If a body travels unequal distances in equal intervals of time, it is said to be in non-uniform motion.
Speed of a body is the distance travelled by the body in unit time.
If a body travels a distance s in time t then its speed v is:
The SI unit of speed is metre per second (m/s or ms-1).
Since speed has only magnitude it is a scalar quantity.
Average speed is the ratio of the total distance travelled to the total time taken.
The rate of motion of a body is more meaningful if we specify its direction of motion along with speed. The quantity which specifies both the direction of motion and speed is velocity.
Velocity of a body is the displacement of the body per unit time.
Since velocity has both magnitude and direction, it is a vector quantity.
Average velocity is the ratio of the total displacement to the total time taken.
Average velocity is also the mean of the initial velocity u and final velocity v.
Speed and velocity have the same units m/s or ms-1.
| Speed | Velocity |
|---|---|
| Distance traveled per unit time | Displacement per unit time |
| Scalar quantity | Vector quantity |
| Always positive | Can be positive, negative or zero |
| \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] | \[ \text{Velocity} = \frac{\text{Displacement}}{\text{Time}} \] |
The rate of change of velocity with time.
\[ \text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time}} = \frac{v - u}{t} \]
Where:
u = initial velocity, v = final velocity, t = time
Important: An object can have zero velocity and still be accelerating (e.g., at the highest point of vertical throw).
Slope: Represents speed
Slope: Represents acceleration
Area under curve: Represents displacement
Three equations that describe motion with constant acceleration:
1. \( v = u + at \)
2. \( s = ut + \frac{1}{2}at^2 \)
3. \( v^2 = u^2 + 2as \)
Where:
u = initial velocity, v = final velocity,
a = acceleration, s = distance, t = time
Remember: These equations are valid only when acceleration is constant.
When an object moves in a circular path with constant speed.
\[ \text{Velocity} = \frac{2\pi r}{T} \]
Where:
r = radius, T = time period for one revolution
Examples: Earth revolving around Sun, electrons around nucleus, merry-go-round.
Example: A car travels 300m in 20s. Calculate its speed.
\[ \text{Speed} = \frac{300}{20} = 15 \text{ m/s} \]
Example: A bus accelerates uniformly from rest to 20 m/s in 10s. Find acceleration.
\[ a = \frac{v - u}{t} = \frac{20 - 0}{10} = 2 \text{ m/s}^2 \]
Example: A bike moving at 10 m/s accelerates at 2 m/s² for 5s. Find final velocity.
\[ v = u + at = 10 + (2 × 5) = 20 \text{ m/s} \]