Linear Motion
Measurement · Linear Motion
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 17 Linear Motion
Lesson objectives
By the end of this topic, you should be able to:
- Define displacement, speed, velocity and acceleration.
- Distinguish between distance and displacement, and between speed and velocity.
- Draw and interpret distance-time and velocity-time graphs.
- Solve problems involving relative speed.
Linear Motion
Linear motion is movement in a straight line. Describing it needs four quantities, and two important distinctions.
a) Distance and displacement
Distance is how far something has travelled. It is a scalar: size only.
Displacement is how far it ends up from the start, and in what direction. It is a vector: size and direction.
Walk 5 km east then 5 km west. The distance is 10 km, but the displacement is zero, because you are back where you began.
b) Speed and velocity
Speed is distance per unit time. A scalar.
Velocity is displacement per unit time. A vector, so it carries a direction.
A car circling a roundabout at a steady 30 km/h has constant speed. Its velocity changes, because its direction keeps changing.
c) Acceleration
Acceleration is the rate of change of velocity.
Its unit is metres per second per second, written m/s².
That reads as: the velocity changes by 5 m/s during every second.
Deceleration is negative acceleration, meaning the object is slowing.
d) The distance-time graph
Distance is plotted up the page, time across.
The gradient gives the speed. A steeper line means faster travel.
A horizontal line means the distance is not changing, so the object has stopped.
Here 150 km is covered in the first two hours. A rest follows during the third hour, then travel resumes.
Speed on any section is the distance covered divided by the time taken for that section.
e) The velocity-time graph
Velocity is plotted up the page, time across. This graph tells you two things.
The gradient gives the acceleration. A rising line means speeding up; a falling line means slowing down.
The area under the graph gives the distance travelled. That is the key fact and the most examined.
Here the object speeds up for 2 seconds, holds 10 m/s for 2 seconds, then slows to rest. The area is a trapezium: ½(2 + 6) × 10 = 40 metres.
f) Telling the two graphs apart
On a distance-time graph, a horizontal line means stopped.
On a velocity-time graph, a horizontal line means constant speed, which is not stopped at all.
Read the axis label before interpreting. Confusing the two is the standard error here.
g) Relative speed
When two objects move along the same line, their relative speed is what matters.
Moving in opposite directions, add the speeds. Two cars approaching at 60 and 40 km/h close at 100 km/h.
Moving in the same direction, subtract them. A car at 60 overtaking one at 40 gains at 20 km/h.
Then use time = distance ÷ relative speed.
h) Where this is used
Journey planning. Overtaking and following distances. Athletics timing. Any physics problem involving motion.
Words to know
- Distance -- the total length travelled, a scalar quantity.
- Displacement -- the straight-line distance from the start, with direction, a vector.
- Velocity -- displacement divided by time; speed with direction.
- Acceleration -- the rate of change of velocity, measured in m/s².
- Relative speed -- the rate at which the gap between two moving objects changes.
:::checkpoint Check yourself
- What is the difference between distance and displacement?
- What does the gradient of a velocity-time graph give?
- What does the area under a velocity-time graph give?
- Two cars approach each other at 70 km/h and 50 km/h. What is their relative speed? :::
Bridge to practice
The exercises begin with the definitions and the scalar-vector distinction, move through the equations of uniform acceleration and both graph types, and finish with relative speed and overtaking problems. For every graph question, state first which graph you are reading, since the meaning of the gradient depends on it.