| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Basic trajectory calculations |
| Difficulty | Standard +0.8 This is a multi-part projectile motion problem requiring coordinate geometry, algebraic manipulation to derive a distance formula, and calculus to find when distance is decreasing. Part (b) involves substantial algebra to reach the given form, and part (c) requires differentiating and solving a quadratic inequality—more demanding than typical M1 questions which often test single techniques. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks |
|---|---|
| (a) \(s_A = 98t - 4.9t^2\) and \(s_B = 24.5t\) | M1 A1 B1 |
| (b) \(d^2 = (4.9h(20 - t))^2 + (24.5t)^2 = 4.9^2(t^2 - 40t + 400) + (5t)^2\) | M1 A1 A1 |
| \(= 24.01(t^2 - 40t + 400 + 25) = 24.01(t^2 - 40t + 425t^2)\) | M1 M1 A1 |
| (c) \(\frac{d}{dt}(d^2) = 24.01(4t^2 - 120t + 850) < 0\) for decreasing function | M1 A1 |
| When \(4t^2 - 120t + 850 = 0\), \(t = 11.5\) or \(t = 18.5\), so range is \(11.5 \leq t \leq 18.5\) | M1 A1 A1 |
| A1 | Total: 15 marks |
**(a)** $s_A = 98t - 4.9t^2$ and $s_B = 24.5t$ | M1 A1 B1 |
**(b)** $d^2 = (4.9h(20 - t))^2 + (24.5t)^2 = 4.9^2(t^2 - 40t + 400) + (5t)^2$ | M1 A1 A1 |
$= 24.01(t^2 - 40t + 400 + 25) = 24.01(t^2 - 40t + 425t^2)$ | M1 M1 A1 |
**(c)** $\frac{d}{dt}(d^2) = 24.01(4t^2 - 120t + 850) < 0$ for decreasing function | M1 A1 |
When $4t^2 - 120t + 850 = 0$, $t = 11.5$ or $t = 18.5$, so range is $11.5 \leq t \leq 18.5$ | M1 A1 A1 |
A1 | **Total: 15 marks**Two stones are projected simultaneously from a point $O$ on horizontal ground. Stone $A$ is thrown vertically upwards with speed $98$ ms$^{-1}$. Stone $B$ is projected along the smooth ground in a straight line at $24.5$ ms$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\item Find the distances of the two stones from $O$ after $t$ seconds, where $0 \leq t \leq 20$. \hfill [3 marks]
\item Show that the distance $d$ m between the two stones after $t$ seconds is given by
$$d^2 = 24.01(t^2 - 40t^2 + 425t^2).$$ \hfill [6 marks]
\item Hence find the range of values of $t$ for which the distance between the stones is decreasing. \hfill [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q7 [15]}}