Linear motion — KCSE Mathematics

KCSE Mathematics · 99 practice questions · 3 syllabus objectives · 3 revision lessons

34 easy32 medium33 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define displacement, speed, velocity and acceleration and distinguish distance/displacement and speed/velocity

Plot and interpret distance-time and velocity-time graphs for linear motion

Define relative speed and solve problems involving relative speed

Revision Notes

Concise lesson notes for Linear motion, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Motion: Key Terms

In linear motion, it is essential to understand the following concepts:

  • Displacement: This is the shortest distance from the initial to the final position of an object, including direction. It can be positive or negative depending on the direction.
  • Distance: This is the total length of the path traveled by an object, regardless of direction. Distance is always positive.
  • Speed: This is the rate at which an object covers distance. It is a scalar quantity and is calculated as speed = distance/time.
  • Velocity: This is the rate of change of displacement. It is a vector quantity and is given by velocity = displacement/time.
  • Acceleration: This is the rate of change of velocity over time, calculated as acceleration = (final velocity - initial velocity)/time.

To distinguish between these terms, remember:

  • Distance vs. Displacement: Distance is path-dependent while displacement is path-independent.
  • Speed vs. Velocity: Speed does not have a direction, while velocity does.

Key points to remember

  • Displacement is the shortest path with direction.
  • Distance is the total path length, always positive.
  • Speed is distance covered over time, a scalar.
  • Velocity is displacement over time, a vector.
  • Acceleration is change in velocity over time.

Worked example

Define distance and displacement.

  • Distance is the total path length traveled, e.g., 10 km.
  • Displacement is the shortest distance from start to end, e.g., 8 km east.

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Lesson 2: Understanding Distance-Time and Velocity-Time Graphs

Objective: Plot and interpret distance-time and velocity-time graphs for linear motion

In linear motion, distance-time graphs and velocity-time graphs are crucial for analyzing movement.

  1. Distance-Time Graphs:

    • The x-axis represents time, while the y-axis represents distance.
    • A straight line indicates constant speed; a steeper slope indicates higher speed.
    • A horizontal line shows stationary motion.
  2. Velocity-Time Graphs:

    • The x-axis represents time, and the y-axis represents velocity.
    • A horizontal line indicates constant velocity; a line sloping upwards shows acceleration.
    • A line sloping downwards indicates deceleration.

Interpreting Graphs:

  • From a distance-time graph, the slope gives the speed.
  • From a velocity-time graph, the area under the line gives the distance traveled.

Understanding these graphs helps in solving problems related to linear motion effectively.

  • Distance-time graphs show distance vs. time relationship.
  • Velocity-time graphs illustrate velocity changes over time.
  • Slope of distance-time graph indicates speed.
  • Area under velocity-time graph represents distance.
  • Horizontal lines indicate constant motion or rest.

Plot a distance-time graph for a car that travels 60 km in 1 hour, then remains stationary for 1 hour. The graph shows a line rising to 60 km, then a horizontal line at 60 km for the next hour.

Lesson 3: Understanding Relative Speed in Linear Motion

Objective: Define relative speed and solve problems involving relative speed

Relative speed refers to the speed of one object as observed from another moving object. It is crucial in solving problems involving two or more objects in motion. The relative speed can be calculated by adding or subtracting the speeds of the objects, depending on their direction.

Key points to remember:

  • Same direction: Subtract the speeds.
  • Opposite direction: Add the speeds.

For example, if two cars are moving in the same direction at speeds of 60 km/h and 40 km/h:

  • The relative speed = 60 km/h - 40 km/h = 20 km/h.

If they are moving towards each other, the relative speed becomes:

  • The relative speed = 60 km/h + 40 km/h = 100 km/h.

By understanding relative speed, you can easily determine how fast one object is moving compared to another, which is essential in many real-life scenarios, such as racing or meeting points.

  • Relative speed is the speed of one object from another's perspective.
  • Add speeds if objects move in opposite directions.
  • Subtract speeds if objects move in the same direction.
  • Use relative speed to solve motion problems effectively.
  • Understanding relative speed aids in real-life applications.

A train travels at 80 km/h, and a car travels at 60 km/h in the same direction. What is their relative speed?

  • Relative speed = 80 km/h - 60 km/h = 20 km/h.

Sample Questions

Read 3 questions and answers free. Sign up to access all 99 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

State how to calculate the time it takes for two cyclists moving towards each other at different speeds to meet, given their initial distance apart. (4 marks)

Answer & marking scheme

Part (a) — 1 mark
Relative speed = Speed of Cyclist 1 + Speed of Cyclist 2 (1 mk)
Part (b) — 3 marks
Time taken = Distance apart / Relative speed (1 mk)
Identify the distance between them at the start (1 mk)
Insert the speeds of both cyclists into the relative speed formula (1 mk)
2
easySHORT ANSWER3 marks

State the definition of relative speed and provide an example of its application in a scenario involving two vehicles. (3 marks)

Answer & marking scheme

Part (a) — 1 mark
Relative speed is the speed of one object as observed from another object (1 mk)
Part (b) — 2 marks
If Vehicle A travels at 60 km/h and Vehicle B at 40 km/h in the same direction, their relative speed is 20 km/h (1 mk)
If they travel towards each other, their relative speed is 100 km/h (60 + 40) (1 mk)
3
easySHORT ANSWER4 marks

Explain how to calculate the average speed of an object using a distance-time graph. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Identify the total distance travelled as represented on the graph. (1 mk)
Identify the total time taken as represented on the graph. (1 mk)
Use the formula average speed = total distance / total time. (1 mk)
State that the average speed is expressed in units of distance per time, e.g., m/s. (1 mk)
4

Describe how to interpret the slope of a distance-time graph for an object moving uniformly. (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Linear motion" cover?

Displacement, velocity, speed, acceleration, relative speed, distance-time and velocity-time graphs

How many practice questions are available for Linear motion?

HighMarks has 99 Linear motion practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Linear motion for the KCSE exam?

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