Question 4 - A-Level Maths

SPS SPS FM Mechanics 2021 January — Question 4 12 marks

Exam Board	SPS
Module	SPS FM Mechanics (SPS FM Mechanics)
Year	2021
Session	January
Marks	12
Topic	Power and driving force
Type	Variable resistance: find k or constants
Difficulty	Standard +0.3 This is a standard Further Maths mechanics question involving power-force-velocity relationships, connected particles, and work-energy principles. Part (a) is routine algebraic manipulation using P=Fv at constant speed. Part (b) requires resolving forces on the car alone after the towbar breaks—straightforward application of F=ma. Part (c) applies work-energy principle to the trailer decelerating up an incline. All parts follow standard textbook methods with no novel insight required, though the multi-part structure and inclusion of variable resistance makes it slightly above average difficulty for A-level.
Spec	3.02d Constant acceleration: SUVAT formulae 3.03d Newton's second law: 2D vectors 6.02l Power and velocity: P = Fv

A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).

Show that \(\lambda = 8\). [4]
Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. Find the acceleration of the car immediately after the towbar breaks. [4]
Use the work-energy principle to find the value of \(d\). [4]

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A car of mass $600$kg pulls a trailer of mass $150$kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude $200$N. At the instant when the speed of the car is $v\text{ms}^{-1}$, the resistance to the motion of the car is modelled as a force of magnitude $(200 + \lambda v)$N, where $\lambda$ is a constant.

When the engine of the car is working at a constant rate of $15$kW, the car is moving at a constant speed of $25\text{ms}^{-1}$.

\begin{enumerate}[label=(\alph*)]
\item Show that $\lambda = 8$.
[4]

\item Later on, the car is pulling the trailer up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin\theta = \frac{1}{15}$.

The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude $200$N at all times. At the instant when the speed of the car is $v\text{ms}^{-1}$, the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude $(200 + 8v)$N.

The engine of the car is again working at a constant rate of $15$kW.

When $v = 10$, the towbar breaks. The trailer comes to instantaneous rest after moving a distance $d$ metres up the road from the point where the towbar broke.

Find the acceleration of the car immediately after the towbar breaks.
[4]

\item Use the work-energy principle to find the value of $d$.
[4]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Mechanics 2021 Q4 [12]}}

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Q1 3 Q2 11 Q3 8 Q4 12 Q5 6 Q6 11