Acceleration as function of displacement (v dv/dx method)

Questions where acceleration is given explicitly as a function of displacement x, requiring use of v dv/dx = a(x) to find velocity as a function of position.

5 questions · Standard +0.8

6.06a Variable force: dv/dt or v*dv/dx methods

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CAIE M2 2005 June Q5

7 marks Standard +0.8

5 The acceleration of a particle moving in a straight line is $( x - 2.4 ) \mathrm { m } \mathrm { s } ^ { - 2 }$ when its displacement from a fixed point $O$ of the line is $x \mathrm {~m}$. The velocity of the particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and it is given that $v = 2.5$ when $x = 0$. Find

an expression for $v$ in terms of $x$,
the minimum value of $v$.

CAIE M2 2007 June Q2

5 marks Standard +0.8

2 A particle starts from rest at $O$ and travels in a straight line. Its acceleration is $( 3 - 2 x ) \mathrm { ms } ^ { - 2 }$, where $x \mathrm {~m}$ is the displacement of the particle from $O$.

Find the value of $x$ for which the velocity of the particle reaches its maximum value.
Find this maximum velocity.

CAIE M2 2011 June Q4

7 marks Standard +0.8

4 A particle $P$ starts from rest at a point $O$ and travels in a straight line. The acceleration of $P$ is $( 15 - 6 x ) \mathrm { m } \mathrm { s } ^ { - 2 }$, where $x \mathrm {~m}$ is the displacement of $P$ from $O$.

Find the value of $x$ for which $P$ reaches its maximum velocity, and calculate this maximum velocity.
Calculate the acceleration of $P$ when it is at instantaneous rest and $x > 0$.

WJEC Further Unit 6 Specimen Q1

14 marks Challenging +1.2

A ball of mass 0.4 kg is thrown vertically upwards from a point $O$ with initial speed $17 \mathrm {~ms} ^ { - 1 }$. When the ball is at a height of $x \mathrm {~m}$ above $O$ and its speed is $v \mathrm {~ms} ^ { - 1 }$, the air resistance acting on the ball has magnitude $0.01 v ^ { 2 } \mathrm {~N}$.
1. Show that, as the ball is ascending, $v$ satisfies the differential equation
$$40 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 392 + v ^ { 2 } \right)$$
Find an expression for $v$ in terms of $x$.
Calculate, correct to two decimal places, the greatest height of the ball.
State, with a reason, whether the speed of the ball when it returns to $O$ is greater than $17 \mathrm {~ms} ^ { - 1 }$, less than $17 \mathrm {~ms} ^ { - 1 }$ or equal to $17 \mathrm {~ms} ^ { - 1 }$.

Edexcel M3 2012 June Q1

9 marks Standard +0.3

A particle $P$ is moving along the positive $x$-axis. At time $t = 0$, $P$ is at the origin $O$. At time $t$ seconds, $P$ is $x$ metres from $O$ and has velocity $v = 2e^{-t}$ m s$^{-1}$ in the direction of $x$ increasing.

Find the acceleration of $P$ in terms of $x$. [3]
Find $x$ in terms of $t$. [6]