| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Projection from elevated point - angle above horizontal |
| Difficulty | Standard +0.3 This is a standard projectile motion problem with projection from an elevated point. It requires resolving initial velocity using the given angle (tan α = 5/12), applying SUVAT equations for vertical motion to find time, then using energy or velocity components to find final speed. The steps are routine and well-practiced in A-level mechanics, though the numerical values require careful calculation. Slightly easier than average due to the straightforward application of standard techniques. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(s = ut + \frac{1}{2}at^2\) vertically or any complete method to give an equation in \(t\) only | M1 | 3.4 |
| \(-70 = 65\sin\alpha \times t - \frac{1}{2} \times g \times t^2\) | A1 | 1.1b |
| M(A)1 | 1.1b | |
| \(t = 7\) (s) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Horizontal velocity component at A \(= 65\cos\alpha\) (60) | B1 | 3.4 |
| Complete method to find vertical velocity component at A | M1 | 3.4 |
| \(65\sin\alpha - g \times 7\) OR \((-25)^2 + 2g \times 70\) (45) | A1ft | 1.1b |
| Sub for trig and square, add and square root: \(60^2 + (-45)^2\) | M1 | 3.1b |
| 75 Accept 80 (m s\(^{-1}\)) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. an approximate value of \(g\) has been used, the dimensions of the stone could affect its motion, spin of the stone, \(g = 10\) instead of \(9.8\) has been used, \(g\) has been assumed to be constant, wind effect, shape of the stone | B1 | 3.5b |
# Question 4
## 4(a)
Use $s = ut + \frac{1}{2}at^2$ vertically or any complete method to give an equation in $t$ only | M1 | 3.4
$-70 = 65\sin\alpha \times t - \frac{1}{2} \times g \times t^2$ | A1 | 1.1b
M(A)1 | 1.1b
$t = 7$ (s) | A1 | 1.1b
**(4)**
### Notes for 4(a):
- M1: Complete method, correct number of terms, condone sign errors and sin/cos confusion
- A1: Correct equation in $t$ only with at most one error
- M(A)1: Correct equation in $t$ only
- N.B. For 'up and down' methods etc, the two A marks are for all the equations that they use, lose a mark for each error.
- A1: cao ($g = 9.8$, $7.1$ or $7.11$) ($g = 9.81$, $7.1$ or $7.12$)
## 4(b)
Horizontal velocity component at A $= 65\cos\alpha$ (60) | B1 | 3.4
Complete method to find vertical velocity component at A | M1 | 3.4
$65\sin\alpha - g \times 7$ OR $(-25)^2 + 2g \times 70$ (45) | A1ft | 1.1b
Sub for trig and square, add and square root: $60^2 + (-45)^2$ | M1 | 3.1b
75 Accept 80 (m s$^{-1}$) | A1 | 1.1b
**(5)**
### Notes for 4(b):
- B1: Seen, including on a diagram.
- M1: Condone sign errors and sin/cos confusion
- A1ft: Correct expression; accept negative of this, follow their $t$
- M1: Sub for trig and use Pythagoras
- A1: cao ($g = 9.8$ or $9.81$, $75$ or $74.8$)
## 4(c)
e.g. an approximate value of $g$ has been used, the dimensions of the stone could affect its motion, spin of the stone, $g = 10$ instead of $9.8$ has been used, $g$ has been assumed to be constant, wind effect, shape of the stone | B1 | 3.5b
**(1)**
### Notes for 4(c):
- B1: B0 if incorrect extras
**(10 marks)**
---
**General notes:**
Allow column vectors throughout this question4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-12_453_990_244_539}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
A small stone is projected with speed $65 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ at the top of a vertical cliff.\\
Point $O$ is 70 m vertically above the point $N$.\\
Point $N$ is on horizontal ground.\\
The stone is projected at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac { 5 } { 12 }$\\
The stone hits the ground at the point $A$, as shown in Figure 3.\\
The stone is modelled as a particle moving freely under gravity.\\
The acceleration due to gravity is modelled as having magnitude $\mathbf { 1 0 m ~ s } \mathbf { m ~ } ^ { \mathbf { - 2 } }$
Using the model,
\begin{enumerate}[label=(\alph*)]
\item find the time taken for the stone to travel from $O$ to $A$,
\item find the speed of the stone at the instant just before it hits the ground at $A$.
One limitation of the model is that it ignores air resistance.
\item State one other limitation of the model that could affect the reliability of your answers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 3 2021 Q4 [10]}}