Question 1 - A-Level Maths

AQA M2 2009 January — Question 1 4 marks

Exam Board	AQA
Module	M2 (Mechanics 2)
Year	2009
Session	January
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Variable acceleration (1D)
Type	Displacement from velocity by integration
Difficulty	Moderate -0.8 This is a straightforward integration question requiring only direct application of standard integration rules (power rule and trigonometric integration) with a simple initial condition. The integration is routine with no problem-solving or conceptual challenges beyond basic calculus mechanics.
Spec	1.08b Integrate x^n: where n != -1 and sums 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)3.02f Non-uniform acceleration: using differentiation and integration

1 A particle moves along a straight line. At time $t$, it has velocity $v$, where $$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$ When $t = 0$, the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time $t$.

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Question 1:

Answer	Marks	Guidance
$s = \int v \, dt$	M1	Attempt to integrate
$s = t^4 - 8\left(\frac{-\cos 2t}{2}\right) + 5t + c$	A1	Correct integration of at least two terms
$= t^4 + 4\cos 2t + 5t + c$	A1	Fully correct integration
When $t=0$, $s=0$: $0 = 0 + 4 + 0 + c$, so $c = -4$	M1	Use of initial condition
$s = t^4 + 4\cos 2t + 5t - 4$	A1	Correct final answer

# Question 1:

| $s = \int v \, dt$ | M1 | Attempt to integrate |
|---|---|---|
| $s = t^4 - 8\left(\frac{-\cos 2t}{2}\right) + 5t + c$ | A1 | Correct integration of at least two terms |
| $= t^4 + 4\cos 2t + 5t + c$ | A1 | Fully correct integration |
| When $t=0$, $s=0$: $0 = 0 + 4 + 0 + c$, so $c = -4$ | M1 | Use of initial condition |
| $s = t^4 + 4\cos 2t + 5t - 4$ | A1 | Correct final answer |

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1 A particle moves along a straight line. At time $t$, it has velocity $v$, where

$$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$

When $t = 0$, the particle is at the origin.\\
Find an expression for the displacement of the particle from the origin at time $t$.

\hfill \mbox{\textit{AQA M2 2009 Q1 [4]}}

This paper (9 questions)

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Q1 4 Q2 9 Q3 12 Q4 9 Q5 9 Q6 7 Q7 7 Q8 7 Q9 11

Answer	Marks	Guidance
\(s = \int v \, dt\)	M1	Attempt to integrate
\(s = t^4 - 8\left(\frac{-\cos 2t}{2}\right) + 5t + c\)	A1	Correct integration of at least two terms
\(= t^4 + 4\cos 2t + 5t + c\)	A1	Fully correct integration
When \(t=0\), \(s=0\): \(0 = 0 + 4 + 0 + c\), so \(c = -4\)	M1	Use of initial condition
\(s = t^4 + 4\cos 2t + 5t - 4\)	A1	Correct final answer