| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Two projectiles meeting - 2D flight |
| Difficulty | Standard +0.3 This is a standard two-projectile problem requiring systematic application of SUVAT equations and projectile formulas. Part (i) is straightforward calculation from given velocity components. Parts (ii) and (iii) involve routine steps: finding times of flight using vertical motion equations, then using range to find the second projectile's parameters. While multi-step, it requires no novel insight—just methodical application of standard M2 techniques. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(30^2 = V_1^2\sin^2\theta_1 - 2\times9.8\times250\) | M1 | \(\frac{1}{2}mV_1^2 = \frac{1}{2}m(50)^2 + m\times9.8\times250\) |
| \(V_1^2\sin^2\theta_1 = 5800\) AEF | A1 | |
| \(V_1\cos\theta_1 = 40\) | B1 | |
| \(V_1 = 86.0\) | A1 | AG |
| \(\theta_1 = 62.3°\) | A1 | AG [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0 = \sqrt{5800}\,t_p - 4.9t_p^2\) | M1 | \(30 = V_1\sin\theta_1 - 9.8t\) |
| \(t_p = 15.5\) | A1 | \(t = 4.71\) |
| \(-\sqrt{5800} = 30 - 9.8t_q\) | M1 | |
| \(t_q = 10.8\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R = 40\times15.5\) | M1 | |
| \(R = 621\) | A1 | (620, 622) |
| \(V_2\cos\theta_2\times10.8 = 621\) | B1 | \(V_2\cos\theta_2 = 57.4\) |
| \(0 = V_2\sin\theta_2\times10.8 - 4.9\times10.8^2\) | M1 | |
| \(V_2\sin\theta_2 = 53.1\) or \(53.0\) | A1 | (52.9, 53.1) |
| Method to find \(V_2\) or \(\theta_2\) | M1 | |
| \(\theta_2 = 42.8°\) | A1 | \(42.6°\) to \(42.9°\) |
| \(V_2 = 78.2\ \text{m s}^{-1}\) or \(78.1\ \text{m s}^{-1}\) | A1 | or \(78.1°\) [8] [17] |
# Question 6(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $30^2 = V_1^2\sin^2\theta_1 - 2\times9.8\times250$ | M1 | $\frac{1}{2}mV_1^2 = \frac{1}{2}m(50)^2 + m\times9.8\times250$ |
| $V_1^2\sin^2\theta_1 = 5800$ AEF | A1 | |
| $V_1\cos\theta_1 = 40$ | B1 | |
| $V_1 = 86.0$ | A1 | **AG** |
| $\theta_1 = 62.3°$ | A1 | **AG** **[5]** |
---
# Question 6(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0 = \sqrt{5800}\,t_p - 4.9t_p^2$ | M1 | $30 = V_1\sin\theta_1 - 9.8t$ |
| $t_p = 15.5$ | A1 | $t = 4.71$ |
| $-\sqrt{5800} = 30 - 9.8t_q$ | M1 | |
| $t_q = 10.8$ | A1 | **[4]** |
---
# Question 6(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = 40\times15.5$ | M1 | |
| $R = 621$ | A1 | (620, 622) |
| $V_2\cos\theta_2\times10.8 = 621$ | B1 | $V_2\cos\theta_2 = 57.4$ |
| $0 = V_2\sin\theta_2\times10.8 - 4.9\times10.8^2$ | M1 | |
| $V_2\sin\theta_2 = 53.1$ or $53.0$ | A1 | (52.9, 53.1) |
| Method to find $V_2$ or $\theta_2$ | M1 | |
| $\theta_2 = 42.8°$ | A1 | $42.6°$ to $42.9°$ |
| $V_2 = 78.2\ \text{m s}^{-1}$ or $78.1\ \text{m s}^{-1}$ | A1 | or $78.1°$ **[8]** **[17]** |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550}
A particle $P$ is projected with speed $V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta _ { 1 }$ from a point $O$ on horizontal ground. When $P$ is vertically above a point $A$ on the ground its height is 250 m and its velocity components are $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ horizontally and $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically upwards (see diagram).\\
(i) Show that $V _ { 1 } = 86.0$ and $\theta _ { 1 } = 62.3 ^ { \circ }$, correct to 3 significant figures.
At the instant when $P$ is vertically above $A$, a second particle $Q$ is projected from $O$ with speed $V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta _ { 2 } . P$ and $Q$ hit the ground at the same time and at the same place.\\
(ii) Calculate the total time of flight of $P$ and the total time of flight of $Q$.\\
(iii) Calculate the range of the particles and hence calculate $V _ { 2 }$ and $\theta _ { 2 }$.
\hfill \mbox{\textit{OCR M2 2010 Q6 [17]}}