11.1 Linear Motion
Linear motion is the motion of an object along a straight path. Learn how distance, displacement, speed, velocity and acceleration describe this motion, and how a ticker timer is used to analyse it.
Learning Objectives
- Define linear motion as the motion of an object in a straight path.
- State the definitions, differences, and S.I. units for distance and displacement.
- State the definitions, formulas, and S.I. units for speed, average speed, velocity, and acceleration.
- Solve daily life numerical problems involving distance, displacement, speed, average speed, velocity, and acceleration.
- Determine displacement, velocity, acceleration, and types of motion using ticker tapes and ticker tape charts.
Comparison of Distance vs. Displacement and Ticker Tape Patterns
Distance follows the actual path, while displacement is the shortest straight line in a specified direction. Ticker tape dot spacing reveals the type of motion.
Short Explanation
Definition
Linear motion is the motion of an object in a straight path, such as a car driving straight, a falling fruit from a tree, or a rolling bowling ball.
When an object moves, the total route it covers is its distance, a scalar value measured in metres. In contrast, its displacement is the straight-line distance from its starting point to its finishing point in a specified direction.
Distance vs displacement. Distance does not depend on direction, but displacement requires both magnitude and direction. For example, a path of \(80\text{ m}\) north and \(80\text{ m}\) west gives a distance of \(160\text{ m}\), but a displacement of \(113.14\text{ m}\) in the northwest direction.
Speed and velocity. The rate of covering distance is called speed, whereas the rate of change of displacement is velocity, which includes direction. When velocity changes over time, the object undergoes acceleration.
Ticker timer. In the laboratory, motion is studied using a ticker timer operating at \(50\text{ Hz}\), meaning each tick represents \(0.02\text{ seconds}\). The spacing between dots on a ticker tape reveals whether an object is travelling at uniform velocity, accelerating, or decelerating.
Physical Quantities of Linear Motion
Definitions, formulas and S.I. units for distance, displacement, speed, velocity and acceleration.
| Quantity | Definition | Formula | S.I. unit |
|---|---|---|---|
| Distance | Total path length travelled by an object | — | \(\text{m}\) |
| Displacement | Shortest path connecting two locations in a specified direction | — | \(\text{m}\) |
| Speed | Rate of change of distance | \(\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}\) | \(\text{m s}^{-1}\) |
| Average speed | Rate of change of total distance travelled | \(\text{Average speed} = \dfrac{\text{Total distance}}{\text{Total time}}\) | \(\text{m s}^{-1}\) |
| Velocity | Rate of change of displacement | \(\text{Velocity} = \dfrac{\text{Displacement}}{\text{Time}}\) | \(\text{m s}^{-1}\) |
| Acceleration | Rate of change of velocity | \(a = \dfrac{v - u}{t}\) | \(\text{m s}^{-2}\) |
Worked Examples
Example 1: Average speed
A runner runs \(100\text{ m}\) in \(12.58\text{ s}\).
\(\text{Average speed} = \dfrac{100\text{ m}}{12.58\text{ s}} = 7.95\text{ m s}^{-1}\)
Example 2: Velocity
A car travels \(800\text{ m}\) from A to B in \(65\text{ s}\), then turns back \(350\text{ m}\) to C in \(30\text{ s}\).
\(\text{Velocity} = \dfrac{450\text{ m}}{95\text{ s}} = 4.74\text{ m s}^{-1}\)
Example 3: Acceleration
A runner accelerates from rest to \(10.44\text{ m s}^{-1}\) in \(9.58\text{ s}\).
\(a = \dfrac{10.44\text{ m s}^{-1} - 0\text{ m s}^{-1}}{9.58\text{ s}} = 1.09\text{ m s}^{-2}\)
Example 4: Distance and displacement
A runner goes \(100\text{ m}\) North, \(150\text{ m}\) East, then \(100\text{ m}\) South in \(60\text{ s}\).
Distance \(= 350\text{ m}\), displacement \(= 150\text{ m}\) to the east.
Analysis of Motion Using a Ticker Timer
A ticker timer vibrates at 50 Hz, producing dots that reveal velocity and acceleration.
Ticker timer operation. A ticker timer vibrates at \(50\text{ vibrations per second } (50\text{ Hz})\). One tick is the time interval between two adjacent dots, so \(1\text{ tick} = \dfrac{1}{50}\text{ s} = 0.02\text{ s}\). Therefore, \(10\text{ ticks} = 10 \times 0.02\text{ s} = 0.2\text{ s}\).
Photogate. A photogate is a timing device used to measure very fast motion or short time intervals accurately, such as free fall, pendulum oscillation, or a trolley on a track.
Velocity from ticker tape. \(\text{Velocity} = \dfrac{\text{Displacement}}{\text{Time taken}} = \dfrac{\text{Length of ticker tape}}{\text{Number of ticks} \times 0.02\text{ s}}\)
Acceleration from a ticker tape chart. Strips of 10 ticks are cut and aligned side by side. The initial velocity \(u\) is the velocity of ticker tape 1, and the final velocity \(v\) is the velocity of ticker tape 5. The time interval is \(t = (5 - 1) \times 0.2\text{ s} = 0.8\text{ s}\). Acceleration is then \(a = \dfrac{v - u}{t}\).
| Dot pattern | Type of motion |
|---|---|
| Distance between dots is constant | Uniform velocity (zero acceleration) |
| Distance between consecutive dots increases uniformly | Uniform acceleration (velocity increases uniformly) |
| Distance between consecutive dots decreases uniformly | Uniform deceleration (velocity decreases uniformly) |
| Distance between consecutive dots is not constant | Non-uniform velocity |
Try to Answer First
Answer in your mind, then press “Check Answer”.
An athlete runs \(200\text{ m}\) along a straight track in \(25\text{ s}\). What is his average speed?
A car moving at \(15\text{ m s}^{-1}\) accelerates uniformly to \(25\text{ m s}^{-1}\) in \(5\text{ s}\). Calculate its acceleration.
A ticker tape has 5 ticks spanning a total length of \(10\text{ cm}\). If the ticker timer operates at \(50\text{ Hz}\), calculate the velocity of the tape.
Common Mistakes
- Confusing distance with displacement by forgetting that displacement requires a specific direction or straight-line measurement.
- Counting dots instead of intervals when calculating time from a ticker tape. \(n\) dots equals \(n - 1\) ticks.
- Miscalculating the time interval in a ticker tape chart by multiplying all 5 strips instead of the interval between the first and last strips: \((5 - 1) \times 0.2\text{ s} = 0.8\text{ s}\).
- Forgetting to write the correct units or confusing speed/velocity \(\text{m s}^{-1}\) with acceleration \(\text{m s}^{-2}\).
Concept Misunderstandings
Speed and velocity are identical.
Speed only tells how fast an object moves, while velocity includes both speed and the direction of motion.
Deceleration is a completely different physical quantity from acceleration.
Deceleration is acceleration with a negative value — a decrease in velocity over time.
Wide spacing on a ticker tape always means acceleration.
Wide constant spacing means high uniform velocity, while increasing spacing indicates acceleration.
Summary
- Linear motion is the motion of an object in a straight path.
- Distance is the total path length travelled, measured in metres; displacement is the shortest straight path in a specified direction.
- Speed is distance divided by time; velocity is displacement divided by time, both in \(\text{m s}^{-1}\).
- Acceleration is the rate of change of velocity, \(a = \dfrac{v - u}{t}\), measured in \(\text{m s}^{-2}\).
- A ticker timer operates at \(50\text{ Hz}\), so \(1\text{ tick} = 0.02\text{ s}\). Constant gap means uniform velocity, increasing gap means uniform acceleration, and decreasing gap means uniform deceleration.
Short Activity
A. Objective Quiz
1 What is the time interval represented by one tick on a ticker tape when using a ticker timer connected to a 50 Hz power supply?
2 A trolley moves along a track such that the distance between consecutive dots on its ticker tape increases uniformly in the direction of motion. What type of motion is experienced by the trolley?
B. Fill in the Blanks
3 The distance of the shortest path that connects two locations in a specified direction is known as .
4 The S.I. unit for acceleration is .
C. Drag and Drop: Match the Term to Its Description
Drag each term to the correct textbook description. On a phone, tap an answer first, then tap a matching box. Tap the Answer Choices area to return a selected item.
C. Match the Term to Its Description
Write the correct term next to each description.
| Description | Term |
|---|---|
| 1. Total path length travelled by an object | |
| 2. Rate of change of displacement | |
| 3. Object moving with uniform velocity | |
| 4. Rate of change of velocity |
Answer choices: Distance • Velocity • Constant dot spacing • Acceleration
