October 2021 Paper 3 Mechanics Q5

5. At time \(t\) seconds, a particle \(P\) has velocity \(\mathbf{v}\ \text{m s}^{-1}\), where

\[\mathbf{v} = 3t^{\frac{1}{2}}\,\mathbf{i} - 2t\,\mathbf{j} \qquad t \gt 0\]

(a) Find the acceleration of \(P\) at time \(t\) seconds, where \(t \gt 0\) (2)

(b) Find the value of \(t\) at the instant when \(P\) is moving in the direction of \(\mathbf{i} - \mathbf{j}\) (3)

At time \(t\) seconds, where \(t \gt 0\), the position vector of \(P\), relative to a fixed origin \(O\), is \(\mathbf{r}\) metres.

When \(t = 1\), \(\mathbf{r} = -\mathbf{j}\)

(d) Find the exact distance of \(P\) from \(O\) at the instant when \(P\) is moving with speed \(10\ \text{m s}^{-1}\) (6)

Mark scheme (a)

Scheme	Marks	AO
Differentiate \(\mathbf{v}\) wrt \(t\)	M1	3.1a
\(\dfrac{3}{2}t^{-\frac{1}{2}}\mathbf{i} - 2\mathbf{j}\) isw	A1	1.1b
	(2)

Allow column vectors throughout this question

M1: Both powers decreasing by 1 (M0 if vector(s) disappear but allow recovery)

A1: cao

Mark scheme (b)

Allow column vectors throughout this question

M1: Complete method, using \(\mathbf{v}\), to obtain an equation in \(t\) only, allow a sign error

DM1: Dependent on M1, solve for \(t\)

A1: cao

Mark scheme (c)

Scheme	Marks	AO
Integrate \(\mathbf{v}\) wrt \(t\)	M1	3.1a
\(\mathbf{r} = 2t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j}\ (+\mathbf{C})\)	A1	1.1b
\(t = 1\), \(\mathbf{r} = -\mathbf{j} \Rightarrow \mathbf{C} = -2\mathbf{i}\) so \(\mathbf{r} = 2t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j} - 2\mathbf{i}\)	A1	2.2a
	(3)

Allow column vectors throughout this question

M1: Both powers increasing by 1 (M0 if vectors disappear but allow recovery)

A1: Correct expression without \(\mathbf{C}\)

A1: cao

Mark scheme (d)

Scheme	Marks	AO
\(\sqrt{\left(3t^{\frac{1}{2}}\right)^2 + (2t)^2} = 10\) or \(\left(3t^{\frac{1}{2}}\right)^2 + (2t)^2 = 10^2\)	M1	2.1
\(9t + 4t^2 = 100\)	M(A)1	1.1b
\(t = 4\)	A1	1.1b
\(\mathbf{r} = 14\mathbf{i} - 16\mathbf{j}\)	M1	1.1b
\(\sqrt{14^2 + (-16)^2}\)	M1	3.1a
\(\sqrt{452}\ \left(2\sqrt{113}\right)\) (m)	A1	1.1b
	(6)
	(14 marks)

Allow column vectors throughout this question

M1: Use of Pythagoras on \(\mathbf{v}\) and 10 to set up equation in \(t\)

M(A)1: Correct 3 term quadratic in \(t\)

A1: cao

M1: Substitute their numerical \(t\) value into their \(\mathbf{r}\)

M1: Use of Pythagoras to find the magnitude of their \(\mathbf{r}\)

A1: cso