When can you use v=u+at?

Only when the acceleration is constant.

How do you find velocity from acceleration a=f(t)?

Integrate with respect to t and use the initial condition to find the constant.

How do you find displacement from velocity as a function of time?

Integrate v with respect to t, using the initial position for the constant.

How do you handle a=f(x)?

Write v\frac{dv}{dx}=f(x), then \frac12v^2=\int f(x)\,dx+c.

Where does a=v\frac{dv}{dx} come from?

The chain rule: \frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}.

How do you solve \frac{dx}{dt}=f(x)?

Separate the variables: \int\frac{1}{f(x)}\,dx=\int dt.

How do you find the distance travelled, not just the displacement?

Split the motion where v=0 and add the magnitudes of each displacement.

What does the sign of v tell you?

The direction of motion along the line.

If v=\frac4x, what is the acceleration?

a=v\frac{dv}{dx}=-\frac{16}{x^3}.

Variable acceleration in a straight lineEdexcel International A Level Further Maths: Flashcards

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What are the three forms of acceleration for straight-line motion?

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What are the three forms of acceleration for straight-line motion?: $a=\frac{dv}{dt}=\frac{d^2x}{dt^2}=v\frac{dv}{dx}$ .
When can you use $v=u+at$ ?: Only when the acceleration is constant.
How do you find velocity from acceleration $a=f(t)$ ?: Integrate with respect to $t$ and use the initial condition to find the constant.
How do you find displacement from velocity as a function of time?: Integrate $v$ with respect to $t$ , using the initial position for the constant.
How do you handle $a=f(x)$ ?: Write $v\frac{dv}{dx}=f(x)$ , then $\frac12v^2=\int f(x)\,dx+c$ .
Where does $a=v\frac{dv}{dx}$ come from?: The chain rule: $\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}$ .
How do you solve $\frac{dx}{dt}=f(x)$ ?: Separate the variables: $\int\frac{1}{f(x)}\,dx=\int dt$ .
How do you find when a particle is instantaneously at rest?: Solve $v=0$ .
How do you find the distance travelled, not just the displacement?: Split the motion where $v=0$ and add the magnitudes of each displacement.
Solve $\frac{dx}{dt}=2x$ with $x=3$ when $t=0$ .: $x=3e^{2t}$ .
What does the sign of $v$ tell you?: The direction of motion along the line.
If $v=\frac4x$ , what is the acceleration?: $a=v\frac{dv}{dx}=-\frac{16}{x^3}$ .

Exam questions on Variable acceleration in a straight line

A particle $P$ moves along the $x$ -axis. At time $t$ seconds, $t\ge0$ , its acceleration is $(6t-12)$ m s $^{-2}$ in the positive $x$ -direction. When $t=0$ , $P$ is at the origin $O$ and has velocity $9$ m s $^{-1}$ .
Find the displacement of $P$ from $O$ when $t=1$ .2 marks
A particle $P$ moves along the $x$ -axis. When $P$ is at distance $x$ metres from the origin $O$ , its acceleration is $(3-x)$ m s $^{-2}$ in the positive $x$ -direction. When $t=0$ , $P$ is at $O$ moving with speed $4$ m s $^{-1}$ in the positive $x$ -direction.
Find the magnitude and direction of the acceleration of $P$ when it is at its greatest distance from $O$ .2 marks
A particle $P$ moves along the positive $x$ -axis. When $P$ is at distance $x$ metres from the origin $O$ , its velocity is $2x$ m s $^{-1}$ in the positive $x$ -direction. When $t=0$ , $x=3$ .
Show that $x=3e^{2t}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).