| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Collision with fixed wall |
| Difficulty | Challenging +1.8 This M4 oblique impact question requires understanding that impulse is perpendicular to the wall, decomposing velocities into components parallel and perpendicular to the wall (direction 2i+j), applying Newton's experimental law with the coefficient of restitution, and calculating energy loss. The multi-step vector geometry and restitution application in an oblique context makes this significantly harder than routine mechanics, though the 'show that' part (a) provides scaffolding for parts (b) and (c). |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.03g Impulse in 2D: vector form6.03k Newton's experimental law: direct impact |
Total
Question 7:
7
Total
PMT
Leave
blank
1. A particle P of mass 0.5 kg falls vertically from rest. After t seconds it has speed v m s–1.
A resisting force of magnitude 1.5v newtons acts on P as it falls.
(a) Show that 3v = 9.8(1 – e–3t).
(8)
(b) Find the distance that P falls in the first two seconds of its motion.
(5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2
*P41819A0228*
PMT
Leave
blank
2. A
2 50 m
m s–1
3
B C
Figure 1
A river is 50 m wide and flows between two straight parallel banks. The river flows with
2
a uniform speed of m s–1 parallel to the banks. The points A and B are on opposite banks
3
of the river and AB is perpendicular to both banks of the river, as shown in Figure 1.
Keith and Ian decide to swim across the river. The speed relative to the water of both
10
swimmers is m s–1.
9
Keith sets out from A and crosses the river in the least possible time, reaching the opposite
bank at the point C. Find
(a) the time taken by Keith to reach C,
(2)
(b) the distance BC.
(2)
Ian sets out from A and swims in a straight line so as to land on the opposite bank at B.
(c) Find the time taken by Ian to reach B.
(4)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4
*P41819A0428*
PMT
Leave
blank
Question 2 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
5
*P41819A0528*
Turn over
PMT
Leave
blank
3.
u A (3m)
r
1.6r
B (2m)
2u
r
Figure 2
Two smooth uniform spheres A and B, of equal radius r, have masses 3m and 2m
respectively. The spheres are moving on a smooth horizontal plane when they collide.
Immediately before the collision they are moving with speeds u and 2u respectively. The
centres of the spheres are moving towards each other along parallel paths at a distance 1.6r
apart, as shown in Figure 2.
1
The coefficient of restitution between the two spheres is .
6
Find, in terms of m and u, the magnitude of the impulse received by B in the collision.
(10)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
8
*P41819A0828*
PMT
Leave
blank
Question 3 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
9
*P41819A0928*
Turn over
PMT
Leave
blank
4.
d
P
x
R(m)
(3m)
Figure 3
A small smooth peg P is fixed at a distance d from a fixed smooth vertical wire. A particle
of mass 3m is attached to one end of a light inextensible string which passes over P. The
particle hangs vertically below P. The other end of the string is attached to a small ring R
of mass m, which is threaded on the wire, as shown in Figure 3.
(a) Show that when R is at a distance x below the level of P the potential energy of the
system is
3mg √ ( x2 + d2) – mgx + constant
(4)
(b) Hence find x, in terms of d, when the system is in equilibrium.
(3)
(c) Determine the stability of the position of equilibrium.
(3)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
12
*P41819A01228*
PMT
Leave
blank
Question 4 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
13
*P41819A01328*
Turn over
PMT
Leave
blank
5. A coastguard ship C is due south of a ship S. Ship S is moving at a constant speed of
12 km h–1 on a bearing of 140 °. Ship C moves in a straight line with constant speed
V km h–1 in order to intercept S.
(a) Find, giving your answer to 3 significant figures, the minimum possible value for V.
(3)
It is now given that V = 14
(b) Find the bearing of the course that C takes to intercept S.
(5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
16
*P41819A01628*
PMT
Leave
blank
Question 5 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
17
*P41819A01728*
Turn over
PMT
Leave
blank
6. A particle P of mass m kg is attached to the end A of a light elastic string AB, of natural
length a metres and modulus of elasticity 9ma newtons. Initially the particle and the string
lie at rest on a smooth horizontal plane with AB = a metres. At time t = 0 the end B of the
string is set in motion and moves at a constant speed U m s–1 in the direction AB. The air
resistance acting on P has magnitude 6mv newtons, where v m s–1 is the speed of P. At
time t seconds, the extension of the string is x metres and the displacement of P from its
initial position is y metres.
Show that, while the string is taut,
(a) x + y = Ut
(2)
d2x dx
+ + =
(b) 6 9x 6U
dt2 dt
(5)
You are given that the general solution of the differential equation in (b) is
2U
x = (A + Bt)Ue–3t +
3
where A and B are arbitrary constants.
(c) Find the value of A and the value of B.
(5)
(d) Find the speed of P at time t seconds.
(2)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
20
*P41819A02028*
PMT
Leave
blank
Question 6 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
21
*P41819A02128*
Turn over
PMT
Leave
blank
7. [In this question i and j are perpendicular unit vectors in a horizontal plane]
A small smooth ball of mass m kg is moving on a smooth horizontal plane and strikes a
fixed smooth vertical wall. The plane and the wall intersect in a straight line which is
parallel to the vector 2i + j. The velocity of the ball immediately before the impact is
bi m s–1, where b is positive. The velocity of the ball immediately after the impact is
a(i + j) m s–1, where a is positive.
(a) Show that the impulse received by the ball when it strikes the wall is parallel to
( – i + 2j).
(1)
Find
(b) the coefficient of restitution between the ball and the wall,
(8)
(c) the fraction of the kinetic energy of the ball that is lost due to the impact.
(3)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
24
*P41819A02428*
PMT
Leave
blank
Question 7 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
25
*P41819A02528*
Turn over[In this question $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane]
A small smooth ball of mass $m$ kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector $2\mathbf{i} + \mathbf{j}$. The velocity of the ball immediately before the impact is $b\mathbf{i} + \mathbf{j}$ m s$^{-1}$, where $b$ is positive. The velocity of the ball immediately after the impact is $a(\mathbf{i} + \mathbf{j})$ m s$^{-1}$, where $a$ is positive.
\begin{enumerate}[label=(\alph*)]
\item Show that the impulse received by the ball when it strikes the wall is parallel to $(-\mathbf{i} + 2\mathbf{j})$.
[1]
\end{enumerate}
Find
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item the coefficient of restitution between the ball and the wall,
[8]
\item the fraction of the kinetic energy of the ball that is lost due to the impact.
[3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2013 Q7 [12]}}