Ch 2 · 4.2 Kinematics of Motion in a Straight Line · A Level Mathematics 9709
Topic 2 of 5 in Mechanics — A Level Mathematics 9709 — 69 lessons.
Introduction
A sprinter's race, a lift between floors, a ball thrown up a cliff: all are motion along a straight line. Kinematics describes that motion (where, how fast, how quickly speeding up) without asking what causes it. You will use graphs, the constant acceleration ("suvat") formulae and calculus. Take g = 10 m s⁻². (Syllabus section 4.2, Paper 4.)
Key ideas — Scalars and vectors in one dimension
Distance and speed are scalars (size only). Displacement s, velocity v and acceleration a are vectors: in one dimension the sign shows the direction, so choose a positive direction and stick to it. A particle that goes 5 m forwards and 5 m back has travelled 10 m but has displacement 0. "Deceleration" means the speed is decreasing.
🧑🏫 Teacher's explanation
Imagine you are walking in a long, straight hallway. If you walk 10 metres forward and then 10 metres back to your starting point, your pedometer (which tracks total movement) says you walked 20 metres. This is 'distance'. However, if someone asks where you are relative to where you started, you are at 0 metres. This is 'displacement'.
📖 Definitions
🔍 Worked example — Standard
A car accelerates from rest with a constant acceleration of 2 m s^-2 for 5 seconds, and then moves at a constant velocity for 10 seconds. Find the total displacement of the car.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Reading the graphs
A straight-line v–t graph means constant acceleration; a horizontal one means constant velocity.
🧑🏫 Teacher's explanation
Think of motion like a road trip. A displacement-time graph is like a GPS log showing exactly where you are at every second. If the line is steep, you are moving fast; if it is flat, you are parked. The 'steepness' or gradient tells you your velocity.
📖 Definitions
🔍 Worked example — Easy
A displacement-time graph for a particle is given by s = t^2 - 4t, where s is displacement in meters and t is time in seconds. Find the velocity of the particle at t = 3.
🔍 Worked example — Standard
A velocity-time graph shows a particle moving with constant acceleration from v = 0 m/s to v = 15 m/s in 5 seconds. Calculate the displacement of the particle during this interval.
🔍 Worked example — Exam-style
A particle moves such that its velocity is given by v = 3t^2 - 6t for 0 ≤ t ≤ 4. Find the total distance traveled by the particle in the interval 0 ≤ t ≤ 4.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Find the acceleration of the particle at t = 3. (b) Find the total distance traveled by the particle between t = 0 and t = 4. [5 marks]
Key ideas — The constant acceleration formulae
When a is constant, with u = initial velocity, v = final velocity, t = time:
🧑🏫 Teacher's explanation
Imagine you are driving a car on a long, straight highway. If you press the accelerator pedal and hold it in a fixed position, your speed increases at a steady, predictable rate. This steady rate of change is what we call 'constant acceleration'. Because the change is steady, we don't need complex calculus to find out how far you've gone; we can use simple algebraic shortcuts.
📖 Definitions
🔍 Worked example — Easy
A car accelerates from rest to a velocity of 20 m s^-1 in 5 s. Calculate the acceleration and the distance travelled.
🔍 Worked example — Standard
A stone is thrown vertically upwards from the ground with an initial velocity of 15 m s^-1. Taking g = 9.8 m s^-2 and ignoring air resistance, find the maximum height reached.
🔍 Worked example — Exam-style
A particle moves in a straight line with constant acceleration. It travels 100 m in 4 s and 250 m in the next 4 s. Find the initial velocity u.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Calculus for variable acceleration
When the acceleration is not constant, the suvat formulae do not apply. Use
🧑🏫 Teacher's explanation
Imagine you are driving a car through a city. If you keep your foot steady on the pedal, your acceleration is constant and you can use simple formulas. But in the real world, you tap the brakes, speed up for a green light, and slow down for a turn. Your acceleration is constantly changing. In these cases, simple algebra fails us, and we need calculus to track your movement.
📖 Definitions
🔍 Worked example — Standard
The acceleration of a particle is a = 6t - 4 m s^-2. Given that the particle starts from rest at t = 0, find its velocity at time t.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — More than one particle
When two particles move at the same time, write an expression for each one's position in terms of the same t, measured from the same point, and set them equal for a meeting or overtaking point. Watch for one particle starting later: its time is t - t_0.
🧑🏫 Teacher's explanation
Imagine you are watching two runners on a track. If Runner A starts at the whistle and Runner B starts 10 seconds later, you cannot simply use the same time variable t for both without being careful. If t is the time since the whistle blew, Runner B has only been running for (t - 10) seconds. To solve problems involving two moving objects, you must establish a 'master clock'—a single time variable that tracks everything from the same starting moment.
📖 Definitions
🔍 Worked example — Easy
Particle A moves from a fixed origin with a velocity of 5 m/s constant. Particle B moves from the same origin with a velocity of 8 m/s constant, but starts 3 s after Particle A. Find the time t (from Particle A's start) when Particle B overtakes Particle A.
🔍 Worked example — Standard
A car travels at a constant speed of 20 m/s. A truck starts from rest at the same origin 4 s later and accelerates at a constant rate of 2 m/s^2. Find the time t (from the car's start) when the truck catches the car.
🔍 Worked example — Exam-style
Particle P moves from origin O with velocity v_P = 3t m/s. Particle Q moves from O with constant velocity v_Q = 12 m/s, but starts 2 s after P. Find the time t (from P's start) when they meet.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: a velocity–time graph
A car accelerates uniformly from rest to 12 m s⁻¹ in 4 s, travels at 12 m s⁻¹ for 10 s, then decelerates uniformly to rest in 6 s. Find the total distance and the deceleration.
🔍 Worked example — Easy
A particle moves along a straight line. Its velocity v at time t is given by v = 3t - 6 for 0 ≤ t ≤ 4. Find the displacement of the particle between t=0 and t=4.
🔍 Worked example — Standard
A car accelerates uniformly from 5 m s^-1 to 25 m s^-1 in 5 s, then maintains this velocity for 15 s. Find the total distance traveled.
🔍 Worked example — Exam-style
A cyclist accelerates uniformly from rest to 12 m s^-1 in 4 s, travels at 12 m s^-1 for 10 s, then decelerates uniformly to rest in 6 s. Find the total distance and the deceleration during the final stage.
⚠️ Common errors
Worked examples — Example 2: vertical motion
A stone is thrown upwards at 15 m s⁻¹ from the top of a cliff 20 m high. Find the greatest height above the cliff top and the time to reach the sea below.
🔍 Worked example — Easy
A stone is thrown upwards at 15 m s^-1 from the top of a cliff 20 m high. Find the greatest height above the cliff top.
🔍 Worked example — Standard
Using the same stone (u = 15, a = -10), find the time taken to reach the sea below the cliff.
🔍 Worked example — Exam-style
A particle moves such that its velocity v at time t is given by v = 20 - 10t. Find the total distance travelled by the particle in the first 3 seconds.
⚠️ Common errors
Worked examples — Example 3: variable acceleration
A particle moves along a line with v = 3t^2 - 12t + 9 m s⁻¹. Find when it is at rest, its acceleration at t = 2, and the distance travelled in the first 3 s.
🔍 Worked example — Easy
A particle moves along a straight line with velocity v = 3t^2 - 12t + 9 m s^-1. Find the times when the particle is at rest.
🔍 Worked example — Standard
A particle moves with velocity v = 3t^2 - 12t + 9 m s^-1. Find its acceleration at t = 2 s.
🔍 Worked example — Exam-style
A particle moves with velocity v = 3t^2 - 12t + 9 m s^-1. Find the total distance travelled in the first 3 seconds.
⚠️ Common errors
Worked examples — Example 4: two particles
Car A starts from rest at P with acceleration 2 m s⁻². At the same instant car B passes P at a constant 10 m s⁻¹ in the same direction. When and where does A catch B?
🔍 Worked example — Easy
A particle moves from rest with a constant acceleration of 3 m s^-2. How far has it travelled after 4 s?
🔍 Worked example — Standard
Car A starts from rest at point P with acceleration 2 m s^-2. Car B passes P at the same instant at a constant speed of 10 m s^-1 in the same direction. When and where does A catch B?
🔍 Worked example — Exam-style
Particle X moves with constant acceleration a. Particle Y moves with constant velocity v. They both start from the same point P at t=0. If X catches Y at time T, show that a = 2v/T.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 10^2 = 30^2 + 2a(400) ⇒ a = -1, so deceleration 1 m s⁻²; t = 20 s. (2) 45 = 5t^2 ⇒ t = 3 s; v = 30 m s⁻¹. (3) v = 3t^2 - 18t + 24 = 3(t-2)(t-4): rest at t = 2, 4; s(2) = 20, s(4) = 16, so distance = 20 + 4 = 24 m. (4) v = 6t - t^2; greatest when a = 0, t = 3: v = 9 m s⁻¹. (5) s = tfrac12(4 + 10)(12) = 84 m.