OCR M2 2010 January — Question 5 12 marks

Exam BoardOCR
ModuleM2 (Mechanics 2)
Year2010
SessionJanuary
Marks12
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Mark schemeDownload PDF ↗
TopicMomentum and Collisions 1
TypeDirect collision with energy loss
DifficultyStandard +0.3 This is a standard M2 collision problem requiring conservation of momentum and the given energy loss condition to form two equations in two unknowns. The algebra is straightforward with integer values, and the method is a textbook exercise that students practice repeatedly. Slightly above average difficulty due to the two-equation system and sign considerations for direction, but well within routine M2 expectations.
Spec6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation
5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.
Question 5:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(16 - 12 = 2x + 3y\)M1
\(4 = 2x + 3y\)A1 aef
\(\frac{1}{2}(2)(8)^2 + \frac{1}{2}(2)(3)(4)^2\) or \(\frac{1}{2}(2)(2)x^2 + \frac{1}{2}(2)(3)y^2\)B1
\(\pm\frac{1}{2}(2)(8^2 - x^2)\) or \(\pm\frac{1}{2}(2)(3)(4^2 - y^2)\)
\(\frac{1}{2}(2)(8)^2 + \frac{1}{2}(2)(3)(4)^2 - \frac{1}{2}(2)x^2 - \frac{1}{2}(2)(3)y^2 = 81\)M1
\(2x^2 + 3y^2 = 14\)A1 aef
Attempt to eliminate x or y from linear and quadratic equationM1
\(15y^2 - 24y - 12 = 0\) or \(10x^2 - 16x - 26 = 0\)A1 aef
Attempt to solve three term quadraticM1
\(x = -1\) (or \(x = 2.6\))A1
\(y = 2\) (or \(y = -2/5\))A1
\(x = -1\) and \(y = 2\) onlyA1
speeds 1, 2 away from each otherA1 [12] 
# Question 5:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $16 - 12 = 2x + 3y$ | M1 | |
| $4 = 2x + 3y$ | A1 | aef |
| $\frac{1}{2}(2)(8)^2 + \frac{1}{2}(2)(3)(4)^2$ or $\frac{1}{2}(2)(2)x^2 + \frac{1}{2}(2)(3)y^2$ | B1 | |
| $\pm\frac{1}{2}(2)(8^2 - x^2)$ or $\pm\frac{1}{2}(2)(3)(4^2 - y^2)$ | | |
| $\frac{1}{2}(2)(8)^2 + \frac{1}{2}(2)(3)(4)^2 - \frac{1}{2}(2)x^2 - \frac{1}{2}(2)(3)y^2 = 81$ | M1 | |
| $2x^2 + 3y^2 = 14$ | A1 | aef |
| Attempt to eliminate x or y from linear and quadratic equation | M1 | |
| $15y^2 - 24y - 12 = 0$ or $10x^2 - 16x - 26 = 0$ | A1 | aef |
| Attempt to solve three term quadratic | M1 | |
| $x = -1$ (or $x = 2.6$) | A1 | |
| $y = 2$ (or $y = -2/5$) | A1 | |
| $x = -1$ and $y = 2$ only | A1 | |
| speeds 1, 2 away from each other | A1 | **[12]** |

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5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.

\hfill \mbox{\textit{OCR M2 2010 Q5 [12]}}
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