| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Motion on a slope |
| Type | Motion up rough slope |
| Difficulty | Standard +0.8 This is a substantial two-part M2 question requiring (i) energy methods or equations of motion with friction on an incline (standard but multi-step with component resolution), and (ii) projectile motion analysis to determine if the ball's parabolic path intersects the roof plane—requiring careful coordinate geometry and inequality checking. The second part demands more sophisticated problem-solving than typical M2 questions, pushing it above average difficulty. |
| Spec | 3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model3.03t Coefficient of friction: F <= mu*R model3.03v Motion on rough surface: including inclined planes6.02d Mechanical energy: KE and PE concepts6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| \(R = 0.2 \times 9.8 \times \cos30° (= 1.70)\) | B1 | |
| \(F = 0.1 \times 9.8 \times \cos30° (= 0.849)\) | FT | FT on their \(R\), but not \(R=0.2g\) |
| \(\frac{1}{2} \times 0.2 \times 11^2 - \frac{1}{2} \times 0.2 \times v^2 = 0.2 \times 9.8 \times 5\sin30 + 5 \times 0.849\) | M1 | Use of conservation of energy |
| A1 | ||
| A1 6 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| \(F + 0.2g\sin30 = \pm0.2a\) | M1 | Use of N2L, 3 terms |
| \(a = \pm9.1\) | A1 | |
| \(v^2 = 11^2 + 2 \times a \times 5\) | M1 | Complete method to find \(v\) |
| \(v = 5.44 \text{ m s}^{-1}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = \frac{5\cos30°}{5.44\cos30°}\) | M1 | time to lateral position over \(C\) |
| \(t = 0.919 \text{ s}\) | A1 | |
| \(u = 5.44\sin30° (= 2.72)\) | B1 | |
| \(s = 2.72 \times 0.919 - 4.9 \times 0.919^2\) | M1 | |
| \(s = -1.6\) (or better) | A1 | Ht dropped |
| Ht drop to \(C = 5\sin30° = 2.5\) m | B1 | |
| Ball does not hit the roof | A1 7 | 13 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x\tan\theta - gx^2\sec^2\theta/2V^2\) | B1 | |
| substitute values | M1 | |
| \(V = 5.44\) \(\theta = 30°\) \(x = 5\cos30°\) | A1 | all 3 correct |
| \(y = 2.5 - 9.8 \times 25 \times 3/4 \times 4/3 / (2 \times 5.44^2)\) | A1 | |
| \(y = -1.6\) (or better) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(u = 5.44\sin30° (= 2.72)\) | B1 | |
| \(-2.5 = 5.44\sin30t - 4.9t^2\) | M1 | |
| A1 | aef | |
| \(t = 1.04\) | A1 | time to position level with \(AC\) |
| \(x = 5.44\cos30 \times 1.04 = 4.9\) (or better) | A1 | |
| Horizontal distance from \(B\) to \(C = 5\cos30 = 4.3\) (or better) | B1 | |
| Ball does not hit the roof | A1 7 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x\tan\theta - gx^2\sec^2\theta/2V^2\) | B1 | |
| substitute values | M1 | |
| \(-2.5 = 0.577x - 0.221x^2\) | A1 | aef |
| Attempt to solve quadratic for \(x\) | M1 | |
| \(x = 4.9\) (or better) | A1 | |
| Horizontal distance from \(B\) to \(C = 5\cos30 = 4.3\) (or better) | B1 | |
| Ball does not hit the roof | A1 7 |
| Answer | Marks | Guidance |
|---|---|---|
| \(u = 5.44\sin30° = 2.72\) | B1 | |
| \(-2.5 = 5.44\sin30t - 4.9t^2\) | M1 | |
| A1 | aef | |
| \(t = 1.0\) (or better) | A1 | time to position level with \(AC\) |
| \(T = \frac{5\cos30°}{5.44\cos30°}\) | M1 | time to lateral position over \(C\) |
| \(T = 0.92\) (or better) | A1 | time to lateral position over \(C\) |
| Ball does not hit the roof | A1 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at equation of trajectory | M1 | |
| \(y = 0.577x - 0.221x^2\) | A1 | |
| \(y = -0.577x\) | B1 | Equation of \(BC\) |
| Solving their quadratic and linear equations to get at least \(x\) or \(y\) | M1 | |
| \(x = 5.2\) (or better) or \(y = -3.0\) (or better) | A1 | |
| Horizontal distance from \(B\) to \(C = 5\cos30 = 4.3\) (or better) | B1 | Must be the one needed for comparison |
| Ball does not hit the roof | A1 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at equation of trajectory | M1 | |
| \(y = 0.577x - 0.221x^2\) | A1 | |
| \(y = -0.577x\) | B1 | |
| Solving their quadratic and linear equations | M1 | |
| \(x = 5.2\) (or better) and \(y = -3.0\) (or better) | A1 | |
| Distance \(= 6.0\) (or better) | B1 | Distance from \(B\) to point of intersection |
| Ball does not hit the roof | A1 7 |
## (i)
$R = 0.2 \times 9.8 \times \cos30° (= 1.70)$ | B1 |
$F = 0.1 \times 9.8 \times \cos30° (= 0.849)$ | FT | FT on their $R$, but not $R=0.2g$
$\frac{1}{2} \times 0.2 \times 11^2 - \frac{1}{2} \times 0.2 \times v^2 = 0.2 \times 9.8 \times 5\sin30 + 5 \times 0.849$ | M1 | Use of conservation of energy
| A1 |
| A1 6 | AG
Or last 4 marks of (i):
$F + 0.2g\sin30 = \pm0.2a$ | M1 | Use of N2L, 3 terms
$a = \pm9.1$ | A1 |
$v^2 = 11^2 + 2 \times a \times 5$ | M1 | Complete method to find $v$
$v = 5.44 \text{ m s}^{-1}$ | A1 |
## (ii)
$t = \frac{5\cos30°}{5.44\cos30°}$ | M1 | time to lateral position over $C$
$t = 0.919 \text{ s}$ | A1 |
$u = 5.44\sin30° (= 2.72)$ | B1 |
$s = 2.72 \times 0.919 - 4.9 \times 0.919^2$ | M1 |
$s = -1.6$ (or better) | A1 | Ht dropped
Ht drop to $C = 5\sin30° = 2.5$ m | B1 |
Ball does not hit the roof | A1 7 | **13**
Or first 5 marks of (ii):
$y = x\tan\theta - gx^2\sec^2\theta/2V^2$ | B1 |
substitute values | M1 |
$V = 5.44$ $\theta = 30°$ $x = 5\cos30°$ | A1 | all 3 correct
$y = 2.5 - 9.8 \times 25 \times 3/4 \times 4/3 / (2 \times 5.44^2)$ | A1 |
$y = -1.6$ (or better) | A1 |
Or (ii):
$u = 5.44\sin30° (= 2.72)$ | B1 |
$-2.5 = 5.44\sin30t - 4.9t^2$ | M1 |
| A1 | aef
$t = 1.04$ | A1 | time to position level with $AC$
$x = 5.44\cos30 \times 1.04 = 4.9$ (or better) | A1 |
Horizontal distance from $B$ to $C = 5\cos30 = 4.3$ (or better) | B1 |
Ball does not hit the roof | A1 7 |
Or (ii):
$y = x\tan\theta - gx^2\sec^2\theta/2V^2$ | B1 |
substitute values | M1 |
$-2.5 = 0.577x - 0.221x^2$ | A1 | aef
Attempt to solve quadratic for $x$ | M1 |
$x = 4.9$ (or better) | A1 |
Horizontal distance from $B$ to $C = 5\cos30 = 4.3$ (or better) | B1 |
Ball does not hit the roof | A1 7 |
Or (ii):
$u = 5.44\sin30° = 2.72$ | B1 |
$-2.5 = 5.44\sin30t - 4.9t^2$ | M1 |
| A1 | aef
$t = 1.0$ (or better) | A1 | time to position level with $AC$
$T = \frac{5\cos30°}{5.44\cos30°}$ | M1 | time to lateral position over $C$
$T = 0.92$ (or better) | A1 | time to lateral position over $C$
Ball does not hit the roof | A1 7 |
Or (iii):
Attempt at equation of trajectory | M1 |
$y = 0.577x - 0.221x^2$ | A1 |
$y = -0.577x$ | B1 | Equation of $BC$
Solving their quadratic and linear equations to get at least $x$ or $y$ | M1 |
$x = 5.2$ (or better) or $y = -3.0$ (or better) | A1 |
Horizontal distance from $B$ to $C = 5\cos30 = 4.3$ (or better) | B1 | Must be the one needed for comparison
Ball does not hit the roof | A1 7 |
Or (ii):
Attempt at equation of trajectory | M1 |
$y = 0.577x - 0.221x^2$ | A1 |
$y = -0.577x$ | B1 |
Solving their quadratic and linear equations | M1 |
$x = 5.2$ (or better) and $y = -3.0$ (or better) | A1 |
Distance $= 6.0$ (or better) | B1 | Distance from $B$ to point of intersection
Ball does not hit the roof | A1 7 |A small ball of mass 0.2 kg is projected with speed $11 \text{ ms}^{-1}$ up a line of greatest slope of a roof from a point $A$ at the bottom of the roof. The ball remains in contact with the roof and moves up the line of greatest slope to the top of the roof at $B$. The roof is rough and the coefficient of friction is $\frac{1}{4}$. The distance $AB$ is 5 m and $AB$ is inclined at $30°$ to the horizontal (see diagram).
\begin{enumerate}[label=(\roman*)]
\item Show that the speed of the ball when it reaches $B$ is $5.44 \text{ ms}^{-1}$, correct to 2 decimal places. [6]
\end{enumerate}
The ball leaves the roof at $B$ and moves freely under gravity. The point $C$ is at the lower edge of the roof. The distance $BC$ is 5 m and $BC$ is inclined at $30°$ to the horizontal.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Determine whether or not the ball hits the roof between $B$ and $C$. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR M2 2010 Q7 [13]}}