Maths › Further Mechanics 2 › Further kinematics: acceleration as a function of x, t or v
Further kinematics: acceleration as a function of x, t or v
Six differential equations, and the whole skill is picking the one that separates. Get the pairing right and the integration is routine.
Builds on Newton's laws with a variable force and Solving differential equations.
IN THIS TOPIC
- Choose the form of the acceleration that separates for a given problem.
- Solve the resulting equation and apply the conditions correctly.
- Interpret limiting behaviour, including terminal speed and total distance.
WHAT YOU PROBABLY THINK
Acceleration is dv/dt, so any kinematics problem can be solved by integrating with respect to time.
Pick the pairing that separates
Acceleration has two equal forms and displacement one, giving three starting points and six useful pairings. Which to use is decided by the variable that appears on the right-hand side:
Choosing dv/dt when the acceleration depends on x leaves three variables in one equation and gets nowhere, which is why the opening claim fails in practice even though it is true in principle. The right question to ask first is always: what does the acceleration depend on?
Once separated, integrate both sides and apply the conditions immediately, before rearranging. If displacement is wanted from a velocity that depends on time, a second integration follows.
WORKED EXAMPLE
Resistance proportional to speed
A particle moving at 10 m/s decelerates at 0.5v m/s². Find its speed after 4 s and the distance it covers in that time.
The acceleration depends on v, and time is asked for, so use dv/dt = −0.5v.
Separating: ∫dv/v = −0.5∫dt gives ln v = −0.5t + c, and v = 10 at t = 0 gives v = 10e^(−0.5t).
At t = 4: v = 10e^(−2) = 1.35 m/s.
Distance = ∫v dt = 20(1 − e^(−2)) = 17.3 m.
The same problem against distance
Change the question from 'after 4 seconds' to 'how far before it stops' and the same physics needs the other pairing. Writing the acceleration as v dv/dx and cancelling the v turns a decaying exponential into a straight line, and the answer arrives in two lines instead of an integration by parts.
The limiting behaviour is worth reading off. Under this resistance the speed approaches zero without ever reaching it in finite time, yet the total distance is finite at 20 m. Under a constant driving force against a resistance that grows with speed, the limit is a terminal speed instead: set the acceleration to zero and solve.
YOUR TURN
How far before it stops
For the same particle, find the total distance it travels before coming to rest, and comment.
Show the working
The acceleration depends on v and distance is wanted, so use v dv/dx = −0.5v.
Cancelling v, which is valid while the particle is moving: dv/dx = −0.5, so v = 10 − 0.5x.
v = 0 gives x = 20 m.
The particle never actually stops in finite time, since v decays exponentially, yet the distance it covers is finite. Both statements are true and neither contradicts the other.
THE EXAM BIT
- State which form of the acceleration you are using and why, before separating anything.
- Apply the initial conditions as soon as you have integrated, not after rearranging.
- Watch for cancelling v: it is valid while the particle moves, and worth saying so.
- For terminal speed, set the acceleration to zero rather than taking a limit.
CHECK YOURSELF
A particle has acceleration 4t m/s² and speed 3 m/s at t = 0. Find its speed at t = 2.
Show a hint
The acceleration depends on t, so integrate with respect to t.
Show the answer
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Match the form of the acceleration to what it depends on: dv/dt for t or v, v dv/dx for x or v.
Separate, integrate, then apply the conditions at once; a terminal speed is found by setting the acceleration to zero.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Choose the form of the acceleration that separates for a given problem.
- Solve the resulting equation and apply the conditions correctly.
- Interpret limiting behaviour, including terminal speed and total distance.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.