| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Topic | Power and driving force |
| Type | Work-energy method on incline |
| Difficulty | Standard +0.3 This is a straightforward work-energy problem requiring application of the work-energy principle on an incline. Part (i) uses KE change minus gravitational PE change to find work done; part (ii) requires finding speed at midpoint (using constant force) then applying P=Fv. Standard mechanics with clear methodology, slightly above average due to the two-part structure and need to handle the incline geometry correctly. |
| Spec | 6.02a Work done: concept and definition6.02l Power and velocity: P = Fv |
**Question 9(i) and 9(ii)**
**(i)**
Gain in KE $= \dfrac{1}{2} \times 800 \times (25^2 - 10^2) = 210\,000\ \mathrm{J}$ — M1 ($\frac{1}{2}mv^2 - \frac{1}{2}mu^2 - mg \times 400\sin 2°$)
Loss in PE $= 800 \times 10 \times 400\sin 2° = 111\,678\ \mathrm{J}$ — A1 (Sign wrong: M1A0)
Work done $= 210\,000 - 111\,768 = \mathbf{98\,322\ \mathrm{J}}\ (= 98.3\ \mathrm{kJ})$ — A1
**[3]**
**(ii)**
$a = \dfrac{25^2 - 10^2}{800} = 0.65625$ or $\dfrac{21}{32}$ — M1A1 (Allow in part (i) only if used in part (ii))
$v^2 = 10^2 + 400 \times 0.65625 \Rightarrow v = 19.0394$ — A1
$F + 800 \times 10 \times \sin 2° = 800 \times 0.65625 \Rightarrow F = 245.8$ — M1A1 (3 terms needed for M1)
$P = Fv = 4680 \Rightarrow$ Power is $\mathbf{4.68\ \mathrm{kW}}$ — A1
**[6]**9 A car of mass 800 kg is descending a straight hill which is inclined at $2 ^ { \circ }$ to the horizontal. The car passes through the points $A$ and $B$ with speeds $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The distance $A B$ is 400 m .\\
(i) Assuming that resistances to motion are negligible, calculate the work done by the car's engine over the distance from $A$ to $B$.\\
(ii) Assuming also that the driving force produced by the car's engine remains constant, calculate the power of the car's engine at the mid-point of $A B$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2015 Q9 [6]}}