Further Mathematics · OCR H245 · Option M · Topic 6.06

Further dynamics and kinematics

Linear motion under a variable force.

1 sub-topic
Taught in Year 13
29 practice questions
1 animated lesson

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Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

6.06aLinear motion under a variable forceY13

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Further dynamics

Not needed for Year 12Medium

Acceleration as a function of displacement · Year 13 · 1 mark

A force on a particle depends on its displacement $x$ , and the velocity is wanted in terms of $x$ . Which form of the acceleration should be used?
(1)

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$\dfrac{\mathrm{d}v}{\mathrm{d}t}$ ${}= \dfrac{\mathrm{d}v}{\mathrm{d}x} \times \dfrac{\mathrm{d}x}{\mathrm{d}t}$ ${}= v\dfrac{\mathrm{d}v}{\mathrm{d}x}$ . Then separate the variables. $D$ : $v\dfrac{\mathrm{d}v}{\mathrm{d}x}$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further dynamics

Not needed for Year 12Hard

Variable force · Year 13 · 6 marks

A particle

P

of mass 0.5 kg moves along the positive

x

-axis. When its displacement from

O

x

m, the only force on

P

\dfrac{6}{(x + 1)^2}

N, directed away from

O

P

passes through

O

with speed 3 m s⁻¹.

(a)
Show that $v^2 = 33 - \dfrac{24}{x + 1}$ .
(4)
(b)
Find the speed of $P$ when $x = 1$ .
(1)
(c)
Explain why the speed of $P$ never exceeds $5.74$ m s⁻¹.
(1)

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Part (a)4 marks

$a = v \mathrm{d}v/\mathrm{d}x$ . $0.5v\frac{\mathrm{d}v}{\mathrm{d}x}$ ${}= \frac{6}{(x + 1)^2}$
Separate and integrate. $\tfrac{1}{2}v^2$ ${}= -\frac{12}{x + 1} + c$
$x = 0, v = 3: \ c = \frac{33}{2}$

Part (b)1 mark

$v = 4.58\text{ m s}^{-1}$

Part (c)1 mark

$\frac{24}{x + 1} > 0 \text{ so } v^2 < 33$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.