| Exam Board | OCR |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | SUVAT in 2D & Gravity |
| Type | Two particles: same start time, different heights |
| Difficulty | Standard +0.8 This is a challenging M1 question requiring careful geometric reasoning to relate the two particles' motions through the constraint that they collide. Students must set up SUVAT equations for both particles, use trigonometry to relate horizontal and vertical distances, and solve a system involving both linear and quadratic terms. The multi-step nature, geometric insight required, and algebraic manipulation place this above average difficulty. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.02h Motion under gravity: vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| ia \(s(P) = 4.9T + 0.5x + 4.9T^2\) \(y(Q) = (0) + 0.5 \times 9.8T^2\) | M1 A1 A1 [3] | \(s=ut+0.5at^2\) used along plane or vertically, with \(u = 4.9\) or 0, and \(a = 4.9\) or 9.8 appropriately Accept use of t or T Allow g in Y(Q) |
| b \((m) \times 4.9 = (m)g\sin\theta\) \(\theta = 30\) | M1* A1 [2] | Allow Cor S0 |
| c \(y(Q)/(s(P) = \sin\theta\) OR \(y(Q) = s(P)\sin\theta\) \(0.5 \times 9.8(2/3)^3 / (4.9 \times 2/3 + 2.45(2/3)^2 = 0.5\) OR \(0.5 \times 9.8T^2 - (4.9T + 2.45T^2) \times \sin30(\theta))\) \(T= 2/3\) AG [3] | M1 D*M1 A1 | Uses appropriate trigonometry to relate distances Verification needs explicit value of sin(θ)(θ)\( D*M1 Ratio of distances considered using cv (30) |
| ii \(v = 4.9 + 4.9 \times 2/3\) OR \(v = (0) + 9.8 \times 2/3\) \(v = 8.17 \text{ ms}^{-1}\) \(w = 9.8 \times 2/3 = 6.53 \text{ ms}^{-1}\) | M1 A1 A1 [3] | Uses \(v = u + at\), with appropriate u, a values once 8.2 6.5 |
| **Answer/Working** | **Marks** | **Guidance** |
|---|---|---|
| **ia** $s(P) = 4.9T + 0.5x + 4.9T^2$ $y(Q) = (0) + 0.5 \times 9.8T^2$ | M1 A1 A1 [3] | $s=ut+0.5at^2$ used along plane or vertically, with $u = 4.9$ or 0, and $a = 4.9$ or 9.8 appropriately Accept use of t or T Allow g in Y(Q) |
| **b** $(m) \times 4.9 = (m)g\sin\theta$ $\theta = 30$ | M1* A1 [2] | Allow Cor S0 | **Sin0 = (0.5 x 9.8T)/(4.9T + 0.5x 4.9T^2)$ gets M1, but in ic. Beware circular argument.** |
| **c** $y(Q)/(s(P) = \sin\theta$ OR $y(Q) = s(P)\sin\theta$ $0.5 \times 9.8(2/3)^3 / (4.9 \times 2/3 + 2.45(2/3)^2 = 0.5$ OR $0.5 \times 9.8T^2 - (4.9T + 2.45T^2) \times \sin30(\theta))$ $T= 2/3$ AG [3] | M1 D*M1 A1 | Uses appropriate trigonometry to relate distances Verification needs explicit value of sin(θ)(θ)$ D*M1 Ratio of distances considered using cv (30) | **This may appear in b)** $0.5 \times 9.8(2/3) = (4.9 \times 2/3 + 2.45(2/3)^2) \times 0.5$ OR $0.5 \times 9.8T^2 = (4.9 + 2.45T) \times \sin30$ |
| **ii** $v = 4.9 + 4.9 \times 2/3$ OR $v = (0) + 9.8 \times 2/3$ $v = 8.17 \text{ ms}^{-1}$ $w = 9.8 \times 2/3 = 6.53 \text{ ms}^{-1}$ | M1 A1 A1 [3] | Uses $v = u + at$, with appropriate u, a values once 8.2 6.5 |5\\
\includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614}\\
$X$ is a point on a smooth plane inclined at $\theta ^ { \circ }$ to the horizontal. $Y$ is a point directly above the line of greatest slope passing through $X$, and $X Y$ is horizontal. A particle $P$ is projected from $X$ with initial speed $4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ down the line of greatest slope, and simultaneously a particle $Q$ is released from rest at $Y$. $P$ moves with acceleration $4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, and $Q$ descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below $Y$ at time $T$ s after being set in motion.
\begin{enumerate}[label=(\roman*)]
\item (a) Express in terms of $T$ the distances travelled by the particles before the collision.\\
(b) Calculate $\theta$.\\
(c) Using the answers to parts (a) and (b), show that $T = \frac { 2 } { 3 }$.
\item Calculate the speeds of the particles immediately before they collide.
\end{enumerate}
\hfill \mbox{\textit{OCR M1 2011 Q5 [11]}}