M4 June 2016 Q5

5. A toy car of mass 0.5 kg is attached to one end \(A\) of a light elastic string \(AB\), of natural length 1.5 m and modulus of elasticity 27 N. Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with \(AB = 1.5\) m. The end \(B\) is moved in a straight horizontal line directly away from the car, with constant speed \(u\) m s\(^{-1}\). At time \(t\) seconds after \(B\) starts to move, the extension of the string is \(x\) metres and the car has moved a distance \(y\) metres. The effect of air resistance on the car can be ignored.

By modelling the car as a particle, show that, while the string remains taut,

(a)

(i) \(x + y = ut\) (2)

(ii) \(\dfrac{\mathrm{d}^2x}{\mathrm{d}t^2} + 36x = 0\) (4)

(b) Hence show that the string becomes slack when \(t = \dfrac{\pi}{6}\) (3)

(d) Find, in terms of \(u\), the distance the car has travelled when it first reaches end \(B\) of the string. (5)

Mark scheme (a)

Scheme	Marks
(i) Dist moved by end \(B = ut\) Dist of end \(B\) from initial position of car \(= 1.5 + ut\) Length of rope \(= 1.5 + x\) \(\therefore 1.5 + ut = 1.5 + x + y\)	M1
\(\Rightarrow x + y = ut\)	A1
	(2)
(ii) \(T = \dfrac{27x}{1.5} = 18x\)	B1
Eqn of motion for car: \(0.5\dfrac{\mathrm{d}^2y}{\mathrm{d}t^2} = 18x\)	M1
\(x + y = ut\qquad -\ddot{x} = \ddot{y}\)	A1
\((\ddot{x} = -36x)\quad \Rightarrow \dfrac{\mathrm{d}^2x}{\mathrm{d}t^2} + 36x = 0\)	A1
	(4)

Notes

A1 (i) Given answer

M1 (ii) Must start out with \(\ddot{y}\)

A1 Correct substitution for \(\ddot{y}\) - must be explained

A1 No errors seen. Given answer as printed

Mark scheme (b)

Scheme	Marks
\(x = a\sin 6t\)	M1
\(x = 0\quad \sin 6t = 0\)	M1
\(6t = \pi\qquad t = \dfrac{\pi}{6}\)	A1
	(3)

Notes

M1 Find the value of \(t\) when \(x = 0\) or substitute \(t = \dfrac{\pi}{6}\)

A1 No errors seen. Given answer

Mark scheme (c)

Scheme	Marks
\(\dot{x} = 6a\cos 6t\)	M1
\(\Rightarrow \dot{y} = u - 6a\cos 6t,\qquad u = 6a\)	M1 (A1)
\(t = \dfrac{\pi}{12}\quad \dot{y} = u - u\cos\dfrac{6\pi}{12} = u\)	A1
	(3)

Notes

M1 Differentiate their \(x\)

M1 (A1) use \(\dot{y} = u - \dot{x}\) and \(t = 0,\ \dot{y} = 0\) or \(t = \dfrac{\pi}{12}\)

Mark scheme (d)

Scheme	Marks
String slack when \(t = \dfrac{\pi}{6}\qquad \dot{y} = u - u\cos\pi\)	M1
\(= 2u\)	A1
Time \(= 1.5\div u = \dfrac{3}{2u}\)	B1
Total distance travelled \(= \left(\dfrac{\pi}{6} + \dfrac{3}{2u}\right)u + 1.5\)	M1
\(= \dfrac{\pi u}{6} + 3\)	A1
	(5)
	(17 marks)

Notes

M1 Find speed of car when string goes slack.

B1 Time to close gap \(= \dfrac{1.5}{2u - u}\)

M1 Distance travelled by \(B\) + 1.5