5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of \(P \mathrm {~W}\) while moving on a straight road which is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = 0.1\). When moving uphill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When moving downhill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is \(R \mathrm {~N}\). Find the values of \(P\) and \(R\).

## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For using $DF = \frac{P}{v}$ for DF up and down |
| | M1 | For applying Newton's 2nd law up and down |
| $\frac{P}{3} - R - 84g \times 0.1 = 84 \times 1.25$ | A1 | |
| $\frac{P}{10} - R + 84g \times 0.1 = 84 \times 1.25$ | A1 | |
| $\left[P\left(\frac{1}{3} - \frac{1}{10}\right) - 168 = 0\right]$ | M1 | For solving equations for $P$ |
| $P = 720$ | A1 | |
| $\left[R = \frac{720}{3} - 84 - 105\right]$ | M1 | For substitution for $P$ to obtain $R$ |
| $R = 51$ | A1 | |
| **Total: 8 marks** | | |

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5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of $P \mathrm {~W}$ while moving on a straight road which is inclined to the horizontal at an angle $\theta$, where $\sin \theta = 0.1$. When moving uphill, the cyclist's acceleration is $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ at an instant when her speed is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When moving downhill, the cyclist's acceleration is $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ at an instant when her speed is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is $R \mathrm {~N}$. Find the values of $P$ and $R$.

\hfill \mbox{\textit{CAIE M1 2015 Q5 [8]}}