Edexcel M4 2014 June — Question 6

Exam BoardEdexcel
ModuleM4 (Mechanics 4)
Year2014
SessionJune
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSecond order differential equations
TypeModeling context with interpretation
DifficultyChallenging +1.3 This is a standard M4 damped harmonic motion problem requiring setup of differential equation from forces, solving a second-order ODE with given damping coefficient, and finding maximum displacement. While it involves multiple steps and careful algebraic manipulation, the techniques are all standard for M4 students who have practiced damped oscillations. The overdamped case (k = 5n/2) is computationally involved but follows textbook methods without requiring novel insight.
Spec4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral6.02h Elastic PE: 1/2 k x^2
\includegraphics{figure_2} A railway truck of mass \(M\) approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring \(PQ\), which is fixed at the end \(P\). The spring has a natural length \(a\) and modulus of elasticity \(Mn^2a\), where \(n\) is a positive constant. At time \(t = 0\), the spring has length \(a\) and the truck strikes the end \(Q\) with speed \(U\). A resistive force whose magnitude is \(Mkv\), where \(v\) is the speed of the truck at time \(t\), and \(k\) is a positive constant, also opposes the motion of the truck. At time \(t\), the truck is in contact with the buffer and the compression of the buffer is \(x\).
  1. Show that, while the truck is compressing the buffer $$\frac{\text{d}^2x}{\text{d}t^2} + k\frac{\text{d}x}{\text{d}t} + n^2x = 0$$ (4)
It is given that \(k = \frac{5n}{2}\)
  1. Find \(x\) in terms of \(U\), \(n\) and \(t\). (7)
  1. Find, in terms of \(U\) and \(n\), the greatest value of \(x\). (5)
Question 6:
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1. A small smooth ball of mass m is falling vertically when it strikes a fixed smooth plane
which is inclined to the horizontal at an angle (cid:302), where 0° < (cid:302) < 45°. Immediately before
striking the plane the ball has speed u. Immediately after striking the plane the ball moves
in a direction which makes an angle of 45° with the plane. The coefficient of restitution
between the ball and the plane is e. Find, in terms of m, u and e, the magnitude of the
impulse of the plane on the ball.
(11)
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2. A ship A is travelling at a constant speed of 30 km h–1 on a bearing of 050°. Another
ship B is travelling at a constant speed of v km h–1 and sets a course to intercept A. At
1400 hours B is 20 km from A and the bearing of A from B is 290°.
(a) Find the least possible value of v.
(3)
Given that v = 32,
(b) find the time at which B intercepts A.
(8)
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3. A small ball of mass m is projected vertically upwards from a point O with speed U. The
ball is subject to air resistance of magnitude mkv, where v is the speed of the ball and k is
a positive constant.
Find, in terms of U, g and k, the maximum height above O reached by the ball.
(8)
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4. A smooth uniform sphere S is moving on a smooth horizontal plane when it collides
obliquely with an identical sphere T which is at rest on the plane. Immediately before the
collision S is moving with speed U in a direction which makes an angle of 60° with the
line joining the centres of the spheres. The coefficient of restitution between the spheres
is e.
(a) Find, in terms of e and U where necessary,
(i) the speed and direction of motion of S immediately after the collision,
(ii) the speed and direction of motion of T immediately after the collision.
(12)
The angle through which the direction of motion of S is deflected is (cid:303)°.
(b) Find
(i) the value of e for which (cid:303) takes the largest possible value,
(ii) the value of (cid:303) in this case.
(3)
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6.
buffer
Figure 2
A railway truck of mass M approaches the end of a straight horizontal track and strikes a
buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled
as a light horizontal spring PQ, which is fixed at the end P. The spring has a natural
length a and modulus of elasticity Mn2a, where n is a postive constant. At time t = 0, the
spring has length a and the truck strikes the end Q with speed U. A resistive force whose
magnitude is Mkv, where v is the speed of the truck at time t, and k is a positive constant,
also opposes the motion of the truck. At time t, the truck is in contact with the buffer and
the compression of the buffer is x.
(a) Show that, while the truck is compressing the buffer
d2 x dx
+ k + n 2 x = 0
dt 2 dt
(4)
5n
It is given that k =
2
(b) Find x in terms of U, n and t.
(7)
(c) Find, in terms of U and n, the greatest value of x.
(5)
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(Total 16 marks)

TOTAL FOR PAPER: 75 MARKS

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Question 6:
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1. A small smooth ball of mass m is falling vertically when it strikes a fixed smooth plane
which is inclined to the horizontal at an angle (cid:302), where 0° < (cid:302) < 45°. Immediately before
striking the plane the ball has speed u. Immediately after striking the plane the ball moves
in a direction which makes an angle of 45° with the plane. The coefficient of restitution
between the ball and the plane is e. Find, in terms of m, u and e, the magnitude of the
impulse of the plane on the ball.
(11)
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2. A ship A is travelling at a constant speed of 30 km h–1 on a bearing of 050°. Another
ship B is travelling at a constant speed of v km h–1 and sets a course to intercept A. At
1400 hours B is 20 km from A and the bearing of A from B is 290°.
(a) Find the least possible value of v.
(3)
Given that v = 32,
(b) find the time at which B intercepts A.
(8)
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3. A small ball of mass m is projected vertically upwards from a point O with speed U. The
ball is subject to air resistance of magnitude mkv, where v is the speed of the ball and k is
a positive constant.
Find, in terms of U, g and k, the maximum height above O reached by the ball.
(8)
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4. A smooth uniform sphere S is moving on a smooth horizontal plane when it collides
obliquely with an identical sphere T which is at rest on the plane. Immediately before the
collision S is moving with speed U in a direction which makes an angle of 60° with the
line joining the centres of the spheres. The coefficient of restitution between the spheres
is e.
(a) Find, in terms of e and U where necessary,
(i) the speed and direction of motion of S immediately after the collision,
(ii) the speed and direction of motion of T immediately after the collision.
(12)
The angle through which the direction of motion of S is deflected is (cid:303)°.
(b) Find
(i) the value of e for which (cid:303) takes the largest possible value,
(ii) the value of (cid:303) in this case.
(3)
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6.
buffer
Figure 2
A railway truck of mass M approaches the end of a straight horizontal track and strikes a
buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled
as a light horizontal spring PQ, which is fixed at the end P. The spring has a natural
length a and modulus of elasticity Mn2a, where n is a postive constant. At time t = 0, the
spring has length a and the truck strikes the end Q with speed U. A resistive force whose
magnitude is Mkv, where v is the speed of the truck at time t, and k is a positive constant,
also opposes the motion of the truck. At time t, the truck is in contact with the buffer and
the compression of the buffer is x.
(a) Show that, while the truck is compressing the buffer
d2 x dx
+ k + n 2 x = 0
dt 2 dt
(4)
5n
It is given that k =
2
(b) Find x in terms of U, n and t.
(7)
(c) Find, in terms of U and n, the greatest value of x.
(5)
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(Total 16 marks)
TOTAL FOR PAPER: 75 MARKS
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\includegraphics{figure_2}

A railway truck of mass $M$ approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring $PQ$, which is fixed at the end $P$. The spring has a natural length $a$ and modulus of elasticity $Mn^2a$, where $n$ is a positive constant. At time $t = 0$, the spring has length $a$ and the truck strikes the end $Q$ with speed $U$. A resistive force whose magnitude is $Mkv$, where $v$ is the speed of the truck at time $t$, and $k$ is a positive constant, also opposes the motion of the truck. At time $t$, the truck is in contact with the buffer and the compression of the buffer is $x$.

\begin{enumerate}[label=(\alph*)]
\item Show that, while the truck is compressing the buffer
$$\frac{\text{d}^2x}{\text{d}t^2} + k\frac{\text{d}x}{\text{d}t} + n^2x = 0$$
(4)
\end{enumerate}

It is given that $k = \frac{5n}{2}$

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find $x$ in terms of $U$, $n$ and $t$.
(7)
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find, in terms of $U$ and $n$, the greatest value of $x$.
(5)
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2014 Q6}}
This paper (6 questions)
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Q1 Q2 Q3 Q4 Q5 Q6