OCR M3 2006 June — Question 2 8 marks

Exam BoardOCR
ModuleM3 (Mechanics 3)
Year2006
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVariable Force
TypeDistance traveled with variable force
DifficultyChallenging +1.2 This is a standard M3 variable force problem requiring Newton's second law with v dv/dx, separation of variables, and integration with partial fractions. While it involves multiple steps and A2-level calculus techniques, it follows a well-established method taught explicitly in M3 courses with no novel problem-solving insight required.
Spec6.06a Variable force: dv/dt or v*dv/dx methods
2 A duck of mass 2 kg is travelling with horizontal speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it lands on a lake. The duck is brought to rest by the action of resistive forces, acting in the direction opposite to the duck's motion and having total magnitude \(\left( 2 v + 3 v ^ { 2 } \right) \mathrm { N }\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the duck. Show that the duck comes to rest after travelling approximately 1.30 m from the point of its initial contact with the surface of the lake.
Question 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
(implied method)M1 For applying Newton's 2nd Law
(implied method)M1 For using \(a = v(dv/dx)\)
\(2v(dv/dx) = -(2v + 3v^2)\)A1
(implied method)M1 For separating variables and attempting to integrate
\(\frac{2}{3}\ln(2 + 3v) = -x \quad (+C)\)A1ft ft absence of minus sign
\(\left[\frac{2}{3}\ln 14 = C\right]\)M1 For using \(v(0) = 4\)
\(\left[\frac{2}{3}\ln 2 = -x + \frac{2}{3}\ln 14\right]\)M1 For attempting to solve \(v(x) = 0\) for \(x\)
Comes to rest after travelling \(1.30\text{m}\)A1 8 
# Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| (implied method) | M1 | For applying Newton's 2nd Law |
| (implied method) | M1 | For using $a = v(dv/dx)$ |
| $2v(dv/dx) = -(2v + 3v^2)$ | A1 | |
| (implied method) | M1 | For separating variables and attempting to integrate |
| $\frac{2}{3}\ln(2 + 3v) = -x \quad (+C)$ | A1ft | ft absence of minus sign |
| $\left[\frac{2}{3}\ln 14 = C\right]$ | M1 | For using $v(0) = 4$ |
| $\left[\frac{2}{3}\ln 2 = -x + \frac{2}{3}\ln 14\right]$ | M1 | For attempting to solve $v(x) = 0$ for $x$ |
| Comes to rest after travelling $1.30\text{m}$ | A1 | 8 | AG |

---
2 A duck of mass 2 kg is travelling with horizontal speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it lands on a lake. The duck is brought to rest by the action of resistive forces, acting in the direction opposite to the duck's motion and having total magnitude $\left( 2 v + 3 v ^ { 2 } \right) \mathrm { N }$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the duck. Show that the duck comes to rest after travelling approximately 1.30 m from the point of its initial contact with the surface of the lake.

\hfill \mbox{\textit{OCR M3 2006 Q2 [8]}}
This paper (6 questions)
View full paper
Q1 6 Q2 8 Q3 9 Q4 11 Q5 12 Q6 12