| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2017 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Basic trajectory calculations |
| Difficulty | Moderate -0.3 This is a standard M2 projectile question requiring routine application of SUVAT equations and trajectory formulas. Part (a) uses range formula to find U, part (b) applies Pythagoras to velocity components, and part (c) solves for times when velocity angle matches a given value. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Vertical motion: Use of \(v = u + at\) | M1 | Correct equation in \(U\), \(t\) |
| \((\uparrow): -U = U - gt\) | A1 | |
| Horizontal motion: Use of \(s = ut\) | M1 | Second equation in \(U\) and their \(t\). e.g. \(\dfrac{U^2}{2g} = U \times \dfrac{20}{U} - \dfrac{g}{2}\left(\dfrac{20}{U}\right)^2\) |
| \((\rightarrow): 3Ut = 120\) | A1ft | Follow their \(t\) provided it matches the value of \(s\) used |
| \(\Rightarrow U = 14\) | A1 | Answer Given. Need supporting evidence e.g. correct linear equation or solution of quadratic in \(U^2\) giving \(U^2 = 20g\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v = \sqrt{U^2 + (3U)^2}\) | M1 | Correct use of Pythagoras' theorem and \(U = 14\) |
| \(v = 14\sqrt{10} = 44\) or \(44.3\) m s\(^{-1}\) | A1 | Max 3 s.f. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\tan\alpha = \dfrac{1}{4} \Rightarrow \dfrac{V}{3U} = \dfrac{1}{4}\) | M1 | Use angle to find vertical component |
| \(\Rightarrow V = \dfrac{3}{4}U\) | A1 | \((10.5\) m s\(^{-1})\) |
| Use of \(v = u + at\) \((\uparrow)\): \(\pm\dfrac{3}{4}U = U - gt\) | M1 | Condone without \(\pm\). Accept complete alternative routes via suvat |
| (correct unsimplified including \(\pm\)) | A1 | Correct unsimplified (including \(\pm\)) |
| \(t_1 = \dfrac{U}{4g} = 0.36\) s, \(\quad t_2 = \dfrac{7U}{4g} = 2.5\) s | A1 | One value correct. Accept \(\dfrac{7}{2g}\) and \(\dfrac{49}{2g}\), but not \(\dfrac{5}{14}\). Decimals to max 3 s.f. |
| (both values correct) | A1 | Both values correct. Apply accuracy penalty only once |
## Question 8(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Vertical motion: Use of $v = u + at$ | M1 | Correct equation in $U$, $t$ |
| $(\uparrow): -U = U - gt$ | A1 | |
| Horizontal motion: Use of $s = ut$ | M1 | Second equation in $U$ and their $t$. e.g. $\dfrac{U^2}{2g} = U \times \dfrac{20}{U} - \dfrac{g}{2}\left(\dfrac{20}{U}\right)^2$ |
| $(\rightarrow): 3Ut = 120$ | A1ft | Follow their $t$ provided it matches the value of $s$ used |
| $\Rightarrow U = 14$ | A1 | **Answer Given**. Need supporting evidence e.g. correct linear equation or solution of quadratic in $U^2$ giving $U^2 = 20g$ |
**Total: 5 marks**
---
## Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v = \sqrt{U^2 + (3U)^2}$ | M1 | Correct use of Pythagoras' theorem and $U = 14$ |
| $v = 14\sqrt{10} = 44$ or $44.3$ m s$^{-1}$ | A1 | Max 3 s.f. |
**Total: 2 marks**
---
## Question 8(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\tan\alpha = \dfrac{1}{4} \Rightarrow \dfrac{V}{3U} = \dfrac{1}{4}$ | M1 | Use angle to find vertical component |
| $\Rightarrow V = \dfrac{3}{4}U$ | A1 | $(10.5$ m s$^{-1})$ |
| Use of $v = u + at$ $(\uparrow)$: $\pm\dfrac{3}{4}U = U - gt$ | M1 | Condone without $\pm$. Accept complete alternative routes via suvat |
| (correct unsimplified including $\pm$) | A1 | Correct unsimplified (including $\pm$) |
| $t_1 = \dfrac{U}{4g} = 0.36$ s, $\quad t_2 = \dfrac{7U}{4g} = 2.5$ s | A1 | One value correct. Accept $\dfrac{7}{2g}$ and $\dfrac{49}{2g}$, but not $\dfrac{5}{14}$. Decimals to max 3 s.f. |
| (both values correct) | A1 | Both values correct. Apply accuracy penalty only once |
**Total: 6 marks**\begin{enumerate}
\item At time $t = 0$ seconds, a golf ball is hit from a point $O$ on horizontal ground. The horizontal and vertical components of the initial velocity of the ball are $3 U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The ball hits the ground at the point $A$, where $O A = 120 \mathrm {~m}$. The ball is modelled as a particle moving freely under gravity.\\
(a) Show that $U = 14$\\
(b) Find the speed of the ball immediately before it hits the ground at $A$.\\
(c) Find the values of $t$ when the ball is moving at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 1 } { 4 }$.\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2017 Q8 [13]}}