Question 7 - A-Level Maths

Pre-U Pre-U 9795/2 2011 June — Question 7 3 marks

Exam Board	Pre-U
Module	Pre-U 9795/2 (Pre-U Further Mathematics Paper 2)
Year	2011
Session	June
Marks	3
Topic	Impulse and momentum (advanced)
Type	Circular motion with collision
Difficulty	Standard +0.3 This is a two-part question combining standard pendulum energy conservation with oblique collision mechanics. Part (i) uses routine energy methods and resolving forces in polar coordinates—standard A-level Further Maths techniques. Part (ii) involves coefficient of restitution and impulse calculations with components, which are textbook applications. While it requires careful component resolution and multiple steps, all techniques are standard with no novel insight required, placing it slightly above average difficulty.
Spec	3.02g Two-dimensional variable acceleration 6.02d Mechanical energy: KE and PE concepts 6.02i Conservation of energy: mechanical energy principle 6.03k Newton's experimental law: direct impact 6.03l Newton's law: oblique impacts

7 \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817} A particle of mass 0.3 kg is attached to one end \(A\) of a light inextensible string of length 1.5 m . The other end \(B\) of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of \(75 ^ { \circ }\) with the vertical (see diagram). Air resistance may be regarded as being negligible.

Show that, at an instant when the string makes an angle of \(40 ^ { \circ }\) with the vertical, the speed of the particle is \(3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772} A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
(a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.
(b) Find the magnitude of the impulse exerted by the wall on the sphere.
Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.

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(i) Uses conservation of energy:

\(0.3 \times 10 \times 1.5(\cos 40° - \cos 75°) = \dfrac{1}{2} \times 0.3v^2\) where speed is \(v\) ms\(^{-1}\).

\(\Rightarrow v = 3.90\) (AG) (\(g = 9.8\) gets M1A1A0) M1A1, A1 [3]

(ii) Transverse component \(= -g\sin 40° = -6.428 = -6.43\) ms\(^{-2}\) (Ignore sign). B1

Radial component \(= \dfrac{v^2}{r} = \dfrac{15.216...}{1.5} = 10.1(4)\) ms\(^{-2}\) B1

Magnitude of acceleration \(= \sqrt{(6.428^2 + 10.14^2)} = 12.0\) ms\(^{-2}\). B1\(\checkmark\) [3]

(i) Uses conservation of energy:

$0.3 \times 10 \times 1.5(\cos 40° - \cos 75°) = \dfrac{1}{2} \times 0.3v^2$ where speed is $v$ ms$^{-1}$.

$\Rightarrow v = 3.90$ (AG) ($g = 9.8$ gets M1A1A0) **M1A1, A1** [3]

(ii) Transverse component $= -g\sin 40° = -6.428 = -6.43$ ms$^{-2}$ (Ignore sign). **B1**

Radial component $= \dfrac{v^2}{r} = \dfrac{15.216...}{1.5} = 10.1(4)$ ms$^{-2}$ **B1**

Magnitude of acceleration $= \sqrt{(6.428^2 + 10.14^2)} = 12.0$ ms$^{-2}$. **B1$\checkmark$** [3]

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7\\
\includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817}

A particle of mass 0.3 kg is attached to one end $A$ of a light inextensible string of length 1.5 m . The other end $B$ of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of $75 ^ { \circ }$ with the vertical (see diagram). Air resistance may be regarded as being negligible.
\begin{enumerate}[label=(\roman*)]
\item Show that, at an instant when the string makes an angle of $40 ^ { \circ }$ with the vertical, the speed of the particle is $3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 3 significant figures.
\item By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle.\\
\includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772}

A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $60 ^ { \circ }$ to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
\item (a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.\\
(b) Find the magnitude of the impulse exerted by the wall on the sphere.
\item Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.
\end{enumerate}

\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2011 Q7 [3]}}

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