| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Deriving trajectory equation |
| Difficulty | Standard +0.3 Part (a) is a standard derivation of the trajectory equation found in all mechanics textbooks, requiring only substitution of parametric equations. Part (b) requires finding when y = 3H/4, solving a quadratic, and using the relationship between H and initial conditions, but follows a predictable problem-solving pattern with tan α given explicitly. This is slightly easier than average due to its routine nature and clear structure. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments3.02i Projectile motion: constant acceleration model |
A particle $P$ is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a time $t$ are denoted by $x$ and $y$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Derive the equation of the trajectory of $P$ in the form
$$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]
\item The greatest height of $P$ above the plane is denoted by $H$. When $P$ is at a height of $\frac{3}{4}H$, it is travelling at a horizontal distance $d$.
Given that $\tan \alpha = 3$ and in terms of $H$, the two possible values of $d$. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q6 [9]}}