Table of Contents
Motion, Distance, Displacement, Speed and Velocity
1. Introduction
Motion is one of the fundamental subjects of mechanics. It describes the change in the position of an object with time relative to a reference point or coordinate system.
A moving car, a falling stone, a rotating fan, a flying aircraft and the revolution of Earth around the Sun are examples of motion.
The branch of mechanics that describes motion without considering the forces responsible for it is called kinematics.
The study of motion is essential for understanding acceleration, force, momentum, energy and the laws of motion.
2. Reference Point and Frame of Reference
Motion is always described relative to a reference point or frame of reference.
A reference point is a position used to determine whether an object has changed its location.
A frame of reference is a coordinate system, together with a method of measuring time, used to describe the position and motion of objects.
For example, a passenger sitting inside a moving bus is at rest relative to the bus but is moving relative to a person standing on the roadside.
Therefore, motion and rest are relative concepts.
2.1 Types of reference frames
Inertial frame: A frame of reference in which Newton’s laws of motion hold without introducing fictitious forces. A frame moving at constant velocity relative to another inertial frame is also inertial.
Non-inertial frame: An accelerating or rotating frame in which fictitious forces may be introduced to apply Newtonian mechanics.
3. Types of Motion
Motion can be classified according to the path followed or the manner in which an object moves.
3.1 Translational motion
An object undergoes translational motion when all its points move through equal displacements over the same time interval, in the same direction.
Examples include:
- A car moving along a straight road.
- A lift moving vertically.
- A train moving along a straight track.
Translational motion may be rectilinear or curvilinear.
3.2 Rotational motion
Rotational motion occurs when an object turns around an axis.
Examples include:
- A rotating ceiling fan.
- A spinning wheel.
- Earth’s rotation around its axis.
3.3 Oscillatory motion
Oscillatory motion is repeated motion to and fro around an equilibrium position.
Examples include:
- A swinging pendulum.
- A vibrating spring.
- A vibrating tuning fork.
3.4 Circular motion
Circular motion occurs when an object follows a circular path around a fixed centre or axis.
Examples include:
- A satellite moving in an approximately circular orbit.
- A stone tied to a string and rotated.
- The movement of the tip of a rotating fan blade.
3.5 Periodic motion
Periodic motion repeats itself after equal intervals of time.
Examples include idealised pendulum motion, the oscillation of a spring and the rotation of Earth’s axis relative to the Sun.
4. Distance
Distance is the total length of the actual path travelled by an object.
It is a scalar quantity because it has magnitude but no direction.
The SI unit of distance is the metre (m).
4.1 Example
A person walks 3 metres east and then 4 metres west.
Total distance travelled:
[
3+4=7\ m
]
Distance is always non-negative and depends on the actual path followed.
5. Displacement
Displacement is the change in the position of an object from its initial point to its final point, measured along the shortest directed path between those positions.
It is a vector quantity because it has both magnitude and direction.
[
\Delta x=x_f-x_i
]
Where:
- (\Delta x) = displacement
- (x_f) = final position
- (x_i) = initial position
5.1 Example
A person walks 3 metres east and then 4 metres west.
If east is taken as positive:
[
\Delta x=3-4=-1\ m
]
The displacement is 1 metre west.
The total distance is 7 metres, while the magnitude of displacement is 1 metre.
5.2 Distance versus displacement
| Distance | Displacement |
|---|---|
| Total path length | Change in position |
| Scalar quantity | Vector quantity |
| Depends on actual path | Depends on initial and final positions |
| Always non-negative | Can be positive, negative or zero in one dimension |
| Cannot be less than displacement magnitude | Magnitude cannot exceed distance |
Important examination point: When an object completes a full circular journey and returns to its starting point, its displacement is zero, although the distance travelled is not zero.
6. Speed
Speed is the rate at which distance is travelled.
Average speed is calculated by dividing total distance by total time.
[
\text{Average Speed}=
\frac{\text{Total Distance}}{\text{Total Time}}
]
The SI unit of speed is metre per second (m/s).
Other common units include kilometres per hour (km/h).
6.1 Conversion of speed units
[
1\ km/h=\frac{5}{18}\ m/s
]
[
1\ m/s=3.6\ km/h
]
To convert km/h into m/s, multiply by 5/18.
To convert m/s into km/h, multiply by 3.6.
6.2 Example
A car travels 150 km in 3 hours.
[
\text{Average Speed}=\frac{150}{3}=50\ km/h
]
Its average speed is 50 km/h.
6.3 Types of speed
Uniform speed: An object covers equal distances in equal intervals of time.
Non-uniform speed: An object covers unequal distances in equal intervals of time.
Instantaneous speed: The speed of an object at a particular instant.
Average speed: The total distance travelled divided by total elapsed time.
7. Velocity
Velocity is the rate of change of displacement with time.
Average velocity is calculated by dividing total displacement by total time.
[
\text{Average Velocity}=
\frac{\text{Displacement}}{\text{Time}}
]
The SI unit is metre per second (m/s).
Velocity is a vector quantity because it has both magnitude and direction.
7.1 Example
A person walks 3 metres east and then 4 metres west in 7 seconds.
The total displacement is 1 metre west.
Taking east as positive:
[
v_{\text{avg}}=\frac{-1}{7}
]
[
v_{\text{avg}}=-0.143\ m/s
]
The average velocity is approximately 0.143 m/s west.
7.2 Types of velocity
Uniform velocity: An object has constant velocity when its speed and direction remain unchanged.
Variable velocity: Velocity changes when either speed or direction changes.
Instantaneous velocity: Velocity at a particular instant.
Average velocity: Total displacement divided by total time.
8. Speed versus Velocity
| Speed | Velocity |
|---|---|
| Rate of change of distance | Rate of change of displacement |
| Scalar quantity | Vector quantity |
| Has magnitude only | Has magnitude and direction |
| Cannot be negative | Can be positive, negative or zero in one dimension |
| Depends on path length | Depends on displacement |
| Average speed uses total distance | Average velocity uses total displacement |
An object moving at constant speed around a circular track has changing velocity because its direction continuously changes.
9. Uniform and Non-Uniform Motion
9.1 Uniform motion
Uniform motion occurs when an object covers equal distances in equal intervals of time.
For example, a train moving 20 metres every second along a straight track at constant velocity is in uniform motion.
9.2 Non-uniform motion
Non-uniform motion occurs when an object covers unequal distances in equal intervals of time or its direction changes.
Examples include:
- A car accelerating in traffic.
- A falling object gaining speed.
- A vehicle travelling around a curved road at constant speed.
10. Graphical Representation of Motion
Graphs are useful for understanding and analysing motion.
10.1 Distance-time graph
A distance-time graph represents distance against elapsed time.
- A straight sloping line indicates constant speed.
- A horizontal line indicates that the object is at rest.
- A steeper line indicates greater speed.
- A curved line indicates changing speed.
The gradient of a distance-time graph represents speed.
[
\text{Speed}=\frac{\text{Change in Distance}}{\text{Change in Time}}
]
10.2 Displacement-time graph
A displacement-time graph represents displacement against time.
Its gradient represents velocity.
- Positive gradient indicates positive velocity.
- Negative gradient indicates negative velocity.
- Zero gradient indicates rest.
- Changing gradient indicates changing velocity.
10.3 Velocity-time graph
A velocity-time graph represents velocity against time.
Its gradient represents acceleration.
The signed area under the graph represents displacement.
For motion in one dimension, positive and negative areas must be considered with their signs.
10.4 Acceleration-time graph
An acceleration-time graph represents acceleration against time.
The area under the graph represents the change in velocity.
11. Circular Motion and Changing Velocity
An object moving along a circular path changes its direction continuously, even if its speed remains constant.
This means that uniform circular motion is accelerated motion.
The acceleration is directed towards the centre of the circular path and is called centripetal acceleration.
[
a_c=\frac{v^2}{r}
]
Where:
- (a_c) = centripetal acceleration
- (v) = tangential speed
- (r) = radius of the circular path
The force responsible for this inward acceleration is called centripetal force.
12. Relative Motion
Relative motion describes the motion of one object as observed from another moving object.
For two objects moving along the same straight line, their relative velocity depends on their directions.
Same direction:
[
v_{AB}=v_A-v_B
]
Opposite directions:
[
v_{AB}=v_A+v_B
]
For example, if two cars move in the same direction at 80 km/h and 60 km/h, the relative speed of the faster car with respect to the slower car is 20 km/h.
If they move in opposite directions, their relative speed is 140 km/h.
13. Worked Examples
Example 1: Average speed
A bus travels 120 km in 2 hours and then 60 km in 1 hour. Calculate its average speed.
Total distance:
[
120+60=180\ km
]
Total time:
[
2+1=3\ hours
]
Average speed:
[
\frac{180}{3}=60\ km/h
]
Example 2: Average velocity
A person moves 100 metres east and then 40 metres west in 20 seconds.
Net displacement:
[
100-40=60\ m\ east
]
Average velocity:
[
\frac{60}{20}=3\ m/s\ east
]
Example 3: Distance and displacement in circular motion
A runner completes one full lap of a circular track with a circumference of 400 metres.
- Distance travelled = 400 m
- Displacement = 0 m
The runner returns to the initial position.
36 Key Takeaways for Exams
- Motion is a change in position with time relative to a reference frame.
- Rest and motion are relative concepts.
- Kinematics studies motion without considering its causes.
- An inertial frame is a reference frame in which Newton’s laws hold without fictitious forces.
- Translational motion involves movement from one position to another.
- Rotational motion occurs around an axis.
- Oscillatory motion involves repeated to-and-fro movement.
- Circular motion follows a circular path.
- Periodic motion repeats after equal intervals of time.
- Distance is the total length of the actual path travelled.
- Distance is a scalar quantity.
- The SI unit of distance is metre.
- Displacement is the change in position of an object.
- Displacement is a vector quantity.
- Displacement can be zero even when distance is not zero.
- The magnitude of displacement cannot exceed distance.
- Speed is the rate of change of distance.
- Average speed equals total distance divided by total time.
- Velocity is the rate of change of displacement.
- Average velocity equals total displacement divided by total time.
- Speed is scalar, whereas velocity is vector.
- Uniform speed means equal distances in equal time intervals.
- Uniform velocity requires constant speed and direction.
- Instantaneous speed refers to speed at a particular instant.
- One metre per second equals 3.6 kilometres per hour.
- One kilometre per hour equals 5/18 metres per second.
- The gradient of a distance-time graph represents speed.
- The gradient of a displacement-time graph represents velocity.
- The gradient of a velocity-time graph represents acceleration.
- The signed area under a velocity-time graph represents displacement.
- The area under an acceleration-time graph represents change in velocity.
- An object moving in a circle at constant speed has changing velocity.
- Centripetal acceleration is directed towards the centre of a circular path.
- Relative motion describes movement from the perspective of another reference frame.
- Relative speed in the same direction is generally the difference between speeds.
- Relative speed in opposite directions is the sum of speeds when both move along the same straight line.