| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Deriving trajectory equation |
| Difficulty | Moderate -0.8 This is a standard projectile motion question requiring routine application of kinematic equations and elimination of the parameter t. All steps are textbook procedures: resolve initial velocity, write parametric equations, eliminate t to get trajectory, find range by setting y=0. No problem-solving insight needed beyond direct formula application. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = (20\cos45)t\) | B1 | Or sin45, \(1/\sqrt{2}\), 0.707 |
| \(y = (20\sin45)t - gt^2/2\) | B1 [2] | Or cos45, \(1/\sqrt{2}\), 0.707 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = (20\sin45)(x/(20\cos45)) - g[x/(20\cos45)]^2/2\) | M1 | Substitutes \(t = x/(20\cos45)\) at least once |
| \(y = x - x^2/40\) AG | A1 [2] | Only from \(g = 10\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = 40\) m | B1 [1] |
## Question 1:
### Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = (20\cos45)t$ | B1 | Or sin45, $1/\sqrt{2}$, 0.707 |
| $y = (20\sin45)t - gt^2/2$ | B1 [2] | Or cos45, $1/\sqrt{2}$, 0.707 |
### Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = (20\sin45)(x/(20\cos45)) - g[x/(20\cos45)]^2/2$ | M1 | Substitutes $t = x/(20\cos45)$ at least once |
| $y = x - x^2/40$ AG | A1 [2] | Only from $g = 10$ |
### Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 40$ m | B1 [1] | |
**Total: [5]**
---1 A small ball is projected with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $45 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground. At time $t \mathrm {~s}$ after projection, the horizontal and vertically upwards displacements of the ball from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.\\
(i) Express $x$ and $y$ in terms of $t$.\\
(ii) Show that the equation of the trajectory of the ball is $y = x - \frac { 1 } { 40 } x ^ { 2 }$.\\
(iii) State the distance from $O$ of the point at which the ball first strikes the ground.
\hfill \mbox{\textit{CAIE M2 2013 Q1 [5]}}