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Rates of change and motion

Interpret the derivative as a rate of change; analyse displacement, velocity and acceleration for motion along a line; integrate a rate to recover the original quantity.

Worked examples

Displacement, velocity and acceleration

Straightforward

Problem

A particle moves along a line. Its displacement at time tt seconds is x(t)=t3−9t2+24t−5x(t) = t^3 - 9t^2 + 24t - 5 metres. (a) Find its velocity and acceleration at t=2t = 2. (b) Find when the particle is at rest. (c) Determine whether the particle is accelerating or decelerating at t=2t = 2.

Recovering displacement from velocity

Moderate

Problem

A particle starts at position x=−2x = -2 m. Its velocity at time tt seconds is v(t)=6t2−4t+1v(t) = 6t^2 - 4t + 1 m/s. Find its displacement at t=3t = 3.

Total distance vs displacement

Challenging

Problem

A particle has velocity v(t)=t2−4v(t) = t^2 - 4 m/s for 0≤t≤40 \leq t \leq 4. Find the total distance travelled.

Practise

Q1·Straightforward
A particle moves along a line so that its displacement at time tt seconds is x(t)=t2−4t+3x(t) = t^2 - 4t + 3 metres. What is the velocity at t=1t = 1?
Q2·Straightforward
A particle has displacement x(t)=t2−4t+3x(t) = t^2 - 4t + 3 metres at time tt seconds. At what time (in seconds) is the particle momentarily at rest?
Q3·Straightforward
A particle has velocity v(t)=3t2−12v(t) = 3t^2 - 12 m/s. Find the acceleration at t=2t = 2 seconds, in m/s².
Q4·Straightforward
A particle moves so that its velocity at time tt is v(t)=2t−6v(t) = 2t - 6 m/s. When is the particle moving in the positive direction?
Q5·Moderate
A particle's displacement is x(t)=t3−6t2+9tx(t) = t^3 - 6t^2 + 9t metres, where t≥0t \geq 0. Find the displacement (in metres) at the first time the particle is at rest.
Q6·Moderate
A particle's velocity is v(t)=6−2tv(t) = 6 - 2t m/s. It starts at position x=3x = 3 m at t=0t = 0. Find its displacement (in metres) at t=4t = 4.
Q7·Moderate
A water tank is being drained. The volume of water (in litres) at time tt minutes is V(t)=500−20t−t2V(t) = 500 - 20t - t^2 for 0≤t≤100 \leq t \leq 10. Find the rate at which water is draining (in litres per minute) at t=3t = 3.
Q8·Moderate
A particle's acceleration is a(t)=6t−4a(t) = 6t - 4 m/s². Its initial velocity is v(0)=5v(0) = 5 m/s. Find the velocity (in m/s) at t=3t = 3.
Q9·Moderate
The population of a town grows at a rate of dPdt=200+30t\dfrac{dP}{dt} = 200 + 30t people per year. If the population is 5000 at t=0t = 0, find the population after 4 years.
Q10·Challenging
A particle moves in a straight line with displacement x(t)=t3−6t2+9tx(t) = t^3 - 6t^2 + 9t metres. Find the total distance (in metres) travelled in the first 3 seconds.
Q11·Challenging
A particle moves so that its velocity is v(t)=t3−6t2+9tv(t) = t^3 - 6t^2 + 9t m/s (t≥0t \geq 0). Find the acceleration (in m/s²) at t=2t = 2.
Q12·Challenging
Oil leaks from a tanker at a rate of R(t)=50e−0.1tR(t) = 50e^{-0.1t} litres per hour, where tt is hours after the leak starts. How many litres leak in the first 10 hours? Give your answer to the nearest whole number.

(Use e−1≈0.3679e^{-1} \approx 0.3679.)