1.0 Introduction to Motion
Everything in the universe is in a state of change. In Physics, we describe this change in position as Motion. However, motion is not an absolute term; it is always described in relation to something else.
Rest vs. Motion
- Rest: An object is said to be at rest if it does not change its position with respect to its immediate surroundings with time.
- Motion: An object is said to be in motion if it changes its position with respect to its immediate surroundings with time.
Reference Point: A fixed point or object with respect to which the state of rest or motion of a body is described.
1.1 Scalars and Vectors
Physical quantities in motion are categorized based on whether they require direction for a complete description.
| Quantity Type | Description | Examples |
|---|---|---|
| Scalar | Has only magnitude (numerical value). | Mass, Time, Distance, Speed |
| Vector | Has both magnitude and direction. | Displacement, Velocity, Force |
1.2 Distance and Displacement
While these terms are used interchangeably in English, they have very specific meanings in Physics.
- Distance: The total path length covered by a moving body. It is a scalar quantity.
- Displacement: The shortest straight-line distance between the initial and final positions of a body. It is a vector quantity.
Displacement can be zero even if the distance covered is not zero. For example, if you run around a circular track and return to the starting point, your distance is the circumference, but your displacement is zero.
An athlete runs 4 km North and then 3 km East. Calculate the total distance and the magnitude of displacement.
Solution:
1. Distance: Total path = $4\,km + 3\,km = 7\,km$.
2. Displacement: Using Pythagoras theorem ($a^2 + b^2 = c^2$):
Calculation: $\sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\,km$.
Final Answer: Distance = $7\,km$; Displacement = $5\,km$ (Northeast).
2.0 Speed and Velocity
To understand how fast an object is moving, we use the terms Speed and Velocity. While they seem similar, the primary difference lies in their relationship with direction.
2.1 Speed
Speed is defined as the distance covered by an object per unit time. It tells us how fast a body is moving, regardless of its direction. It is a scalar quantity.
Speed Formula
$$Speed = \frac{Distance}{Time}$$
SI Unit: $m/s$ | Commercial Unit: $km/h$
2.2 Velocity
Velocity is the displacement of an object per unit time. In other words, it is speed in a specific direction. Since it involves displacement, it is a vector quantity.
Velocity Formula
$$Velocity = \frac{Displacement}{Time}$$
SI Unit: $m/s$ (directed towards a specific point)
To convert $km/h$ to $m/s$ quickly, multiply the value by $\mathbf{\frac{5}{18}}$. To go from $m/s$ to $km/h$, multiply by $\mathbf{\frac{18}{5}}$.
$18\,km/h = 5\,m/s$
A car travels a distance of 180 km in 3 hours. Calculate its speed in $m/s$.
Solution:
1. Calculate Speed in $km/h$: $Speed = \frac{180\,km}{3\,h} = 60\,km/h$
2. Convert to $m/s$: Multiply by $\frac{5}{18}$
3. Calculation: $60 \times \frac{5}{18} = \frac{300}{18} \approx 16.67\,m/s$
Final Answer: The speed is $16.67\,m/s$.
The Speedometer of a car shows the "Instantaneous Speed"—the speed at that exact moment. The Odometer measures the total distance covered by the vehicle!
3.0 Uniform and Non-Uniform Motion
In the real world, objects rarely move at the exact same speed throughout their journey. Based on how an object covers distance over time, we classify motion into two types: Uniform and Non-Uniform.
3.1 Uniform Motion
A body is said to be in Uniform Motion if it covers equal distances in equal intervals of time, no matter how small these time intervals may be. In this state, the speed of the object remains constant.
Example: A car moving on a straight highway at a constant speed of 60 km/h.
3.2 Non-Uniform Motion
A body is said to be in Non-Uniform Motion if it covers unequal distances in equal intervals of time. In this case, the speed of the object keeps changing.
Example: A train starting from a station or a car moving through heavy city traffic.
3.3 Average Speed
For objects in non-uniform motion, we describe their journey using Average Speed. It gives us a single value that represents the entire trip.
Average Speed Formula
$$Average\,Speed = \frac{Total\,Distance\,Travelled}{Total\,Time\,Taken}$$
Average speed is NOT the simple average of two speeds (i.e., $\frac{v_1 + v_2}{2}$). It must always be calculated using the total distance divided by the total time.
A cyclist travels the first 15 km in 1 hour and the next 25 km in 2 hours. Calculate the average speed of the cyclist.
Solution:
1. Total Distance: $15\,km + 25\,km = 40\,km$
2. Total Time: $1\,h + 2\,h = 3\,h$
3. Apply Formula: $Avg.\,Speed = \frac{40\,km}{3\,h}$
4. Calculation: $40 \div 3 = 13.33\,km/h$
Final Answer: The average speed is $13.33\,km/h$.
When a body moves in a circular path with constant speed, its motion is uniform in terms of speed, but non-uniform in terms of velocity because its direction is constantly changing!
4.0 Acceleration
When you are in a car and the driver presses the gas pedal, you feel a "push" as the car picks up speed. In Physics, this change in velocity over time is called Acceleration. It describes how quickly an object is speeding up or slowing down.
What is Acceleration?
Acceleration is defined as the rate of change of velocity with time. Since velocity is a vector, acceleration is also a vector quantity.
SI Unit: $m/s^2$ (metre per second squared)
Acceleration Formula
$$a = \frac{v - u}{t}$$
Where: $a$ = acceleration, $v$ = final velocity, $u$ = initial velocity, $t$ = time taken
4.1 Types of Acceleration
- Positive Acceleration: When the velocity of an object increases with time (e.g., a ball rolling down an inclined plane).
- Negative Acceleration (Retardation): When the velocity of an object decreases with time (e.g., applying brakes to a moving car). This is also known as Deceleration.
- Zero Acceleration: When an object moves with constant velocity (Uniform Velocity).
If you calculate acceleration and get a negative value (e.g., $-2\,m/s^2$), it means the body is slowing down. When writing the value for Retardation, you don't use the negative sign; you simply say "The retardation is $2\,m/s^2$."
A scooter moving at 10 m/s speeds up to 25 m/s in 5 seconds. Find its acceleration.
Solution:
1. Initial Velocity ($u$): $10\,m/s$
2. Final Velocity ($v$): $25\,m/s$
3. Time ($t$): $5\,s$
4. Apply Formula: $a = \frac{v - u}{t} = \frac{25 - 10}{5}$
5. Calculation: $a = \frac{15}{5} = 3\,m/s^2$
Final Answer: The acceleration is $3\,m/s^2$.
Gravity is a natural accelerator! Any object falling freely towards the Earth accelerates at approximately $9.8\,m/s^2$. This means every second, its speed increases by about $9.8\,m/s$.
5.0 Equations of Motion
For a body moving with uniform acceleration in a straight line, there are three fundamental equations that relate initial velocity, final velocity, acceleration, time, and distance covered. These are known as the Equations of Motion.
The Three Equations
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 | $t$ = time taken
$s$ = distance (displacement)
5.1 Graphical Representation
Graphs are a powerful tool to visualize motion. In ICSE Class 7, we focus on two primary types of graphs:
- Distance-Time Graph ($s-t$):
- A straight line passing through the origin indicates uniform speed.
- The slope of this graph gives the speed of the object.
- Velocity-Time Graph ($v-t$):
- A horizontal line (parallel to time axis) indicates constant velocity.
- The slope of this graph gives the acceleration.
- The area under the curve gives the distance travelled.
- If a body starts from rest, take $u = 0$.
- If a body comes to a stop (brakes applied), take $v = 0$.
- For retardation, use a negative sign for $a$ in the equations.
A bus starting from rest moves with a uniform acceleration of $0.1\,m/s^2$ for 2 minutes. Find (a) the speed acquired and (b) the distance travelled.
Solution:
1. Given: $u = 0$, $a = 0.1\,m/s^2$, $t = 2\,min = 120\,s$.
2. (a) Final Speed ($v$):
Using $v = u + at$
$v = 0 + (0.1 \times 120) = 12\,m/s$.
3. (b) Distance ($s$):
Using $s = ut + \frac{1}{2}at^2$
$s = (0 \times 120) + \frac{1}{2}(0.1 \times 120^2)$
$s = 0 + \frac{1}{2}(0.1 \times 14400) = 720\,m$.
Final Answer: Speed acquired is $12\,m/s$ and distance travelled is $720\,m$.
These equations were first formulated by Galileo Galilei. They form the basis of "Kinematics," which is the study of motion without considering the forces that cause it!