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Velocity, acceleration and calculus

Express acceleration in all three equivalent forms: dvdt\frac{dv}{dt}, vdvdxv\frac{dv}{dx}, and ddx ⁣(12v2)\frac{d}{dx}\!\left(\frac{1}{2}v^2\right); solve motion problems by integrating equations of motion with appropriate initial conditions; choose the most efficient form for a given problem.

Worked examples

Choosing the right form of acceleration

Straightforward

Problem

A particle moves along the xx-axis with v=4x3v = 4x^3 m/s. Find the acceleration when x=1x = 1 m.

Finding velocity from acceleration using the energy form

Moderate

Problem

A particle starts from rest at x=0x = 0. Its acceleration is a=3−xa = 3 - x m/s² (for x≥0x \geq 0). Find the speed when x=2x = 2 m.

Solving a separable ODE for velocity as a function of time

Challenging

Problem

A particle moves with acceleration a=5−va = 5 - v m/s². It starts from rest. Find vv as a function of tt, and find the time to reach v=3v = 3 m/s.

Practise

Q1·Straightforward
A particle moves along a straight line so that its velocity at time tt seconds is v=3t2−2tv = 3t^2 - 2t m/s. Find the acceleration of the particle at t=3t = 3 s.
Q2·Straightforward
A particle moves with acceleration a=6t+4a = 6t + 4 m/s². Given that v=2v = 2 m/s when t=0t = 0, find the velocity when t=3t = 3 s.
Q3·Straightforward
A particle starts from rest at the origin with acceleration a=2ta = 2t m/s². Find its displacement (in metres) from the origin when t=4t = 4 s. Give your answer to two decimal places.
Q4·Straightforward
A particle moves so that v=3x2v = 3x^2 m/s, where xx is displacement in metres. Find the acceleration when x=2x = 2 m, using a=vdvdxa = v\dfrac{dv}{dx}.
Q5·Moderate
A particle moves with acceleration a=2xa = 2x m/s² (where xx is displacement). Given that v=0v = 0 when x=0x = 0, find the speed when x=3x = 3 m. Give your answer to two decimal places.
Q6·Moderate
A particle has acceleration a=−4xa = -4x m/s² and starts with v=6v = 6 m/s at x=0x = 0. Find its speed when x=1x = 1 m. Give your answer to two decimal places.
Q7·Moderate
A particle starts from rest at x=1x = 1 with acceleration a=2x2a = \dfrac{2}{x^2} m/s². Find its speed when x=2x = 2 m. Give your answer to two decimal places.
Q8·Moderate
A particle has displacement x=t3−6t2+9tx = t^3 - 6t^2 + 9t m for t≥0t \geq 0. Find the total distance travelled from t=0t = 0 to t=3t = 3 s.
Q9·Moderate
A particle has acceleration a(v)=6−2va(v) = 6 - 2v. Find the terminal velocity (the limiting velocity as t→∞t \to \infty).
Q10·Challenging
A particle moves from rest with acceleration a=4−2va = 4 - 2v m/s² (where a>0a > 0 initially). Find the time (in seconds) taken to reach v=1v = 1 m/s. Give your answer to two decimal places.
Q11·Challenging
A particle has acceleration a=v2a = v^2 m/s² and initial velocity v=1v = 1 m/s at t=0t = 0. Find the velocity when t=0.5t = 0.5 s.
Q12·Straightforward
A particle has acceleration a=e−ta = e^{-t} m/s² and starts from rest at t=0t = 0. Find the velocity (in m/s) at t=2t = 2 s. Give your answer to two decimal places.
Q13·Challenging
Prove that the acceleration of a particle moving in a straight line can be written as a=vdvdx=ddx ⁣(12v2)a = v\dfrac{dv}{dx} = \dfrac{d}{dx}\!\left(\dfrac{1}{2}v^2\right), where vv is velocity and xx is displacement.

✎ Work this one through on paper — proofs are self-assessed.