| Exam Board | OCR |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2015 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Collision or meeting problems |
| Difficulty | Standard +0.3 This is a straightforward M1 kinematics question requiring integration of acceleration to find velocity and position, followed by a simple collision calculation using conservation of momentum. All steps are routine: integrate a(t) = 4 + 12t with given initial conditions, evaluate at t=3, equate positions to find k, and apply momentum conservation with equal masses. No novel insight required, just systematic application of standard techniques. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(v = \int 4 + 12t\ dt\) | M1* | Integrates acceleration; Must see one term correct |
| \(v = 4t + 12t^2/2\ (+c)\) | A1 | Award without \((+c)\) |
| \((t=0, v=2)\ c=2\) and \(v(3) = 4\times3 + 12\times3^2/2\ (+2)\) | D*M1 | Evaluates constant |
| \(v = 68\) m s\(^{-1}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int 4t + 6t^2(+2)dt\) | M1* | Integrates velocity |
| \(x = 4t^2/2 + 6t^3/3 + 2t\ (+d)\) | A1ft | accept omission of \(d\) for all subsequent marks; ft on incorrect (non-zero) \(c\) from (i) |
| \(x(3) = 4\times3^2/2 + 6\times3^3/3\ (+3\times2)\) | D*M1 | |
| \(x = 78\) m | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k = 132\) | B1ft | ft cv(78) \(+ 54\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(v = d(k - 2t^3)/dt\) | M1* | Differentiates displacement; Award even if \(k\) wrong earlier |
| \(v = -2 \times 3t^2\) | A1 | |
| \(v(3) = -6\times3^2\ (= -54)\) | D*M1 | Substitutes \(t=3\) |
| \(68m - 54m = 2mv\) | M1 | Conservation of momentum, must have \(2m\), cv(68); No marks if \(g\) included, even if apparently cancelled |
| \(v = 7\) m s\(^{-1}\) | A1 |
## Question 6:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $v = \int 4 + 12t\ dt$ | M1* | Integrates acceleration; Must see one term correct |
| $v = 4t + 12t^2/2\ (+c)$ | A1 | Award without $(+c)$ |
| $(t=0, v=2)\ c=2$ and $v(3) = 4\times3 + 12\times3^2/2\ (+2)$ | D*M1 | Evaluates constant |
| $v = 68$ m s$^{-1}$ | A1 | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int 4t + 6t^2(+2)dt$ | M1* | Integrates velocity |
| $x = 4t^2/2 + 6t^3/3 + 2t\ (+d)$ | A1ft | accept omission of $d$ for all subsequent marks; ft on incorrect (non-zero) $c$ from (i) |
| $x(3) = 4\times3^2/2 + 6\times3^3/3\ (+3\times2)$ | D*M1 | |
| $x = 78$ m | A1 | |
### Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k = 132$ | B1ft | ft cv(78) $+ 54$ |
### Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $v = d(k - 2t^3)/dt$ | M1* | Differentiates displacement; Award even if $k$ wrong earlier |
| $v = -2 \times 3t^2$ | A1 | |
| $v(3) = -6\times3^2\ (= -54)$ | D*M1 | Substitutes $t=3$ |
| $68m - 54m = 2mv$ | M1 | Conservation of momentum, must have $2m$, cv(68); No marks if $g$ included, even if apparently cancelled |
| $v = 7$ m s$^{-1}$ | A1 | |
---6 A particle $P$ moves in a straight line on a horizontal surface. $P$ passes through a fixed point $O$ on the line with velocity $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t \mathrm {~s}$ after passing through $O$, the acceleration of $P$ is $( 4 + 12 t ) \mathrm { m } \mathrm { s } ^ { - 2 }$.\\
(i) Calculate the velocity of $P$ when $t = 3$.\\
(ii) Find the distance $O P$ when $t = 3$.
A second particle $Q$, having the same mass as $P$, moves along the same straight line. The displacement of $Q$ from $O$ is $\left( k - 2 t ^ { 3 } \right) \mathrm { m }$, where $k$ is a constant. When $t = 3$ the particles collide and coalesce.\\
(iii) Find the value of $k$.\\
(iv) Find the common velocity of the particles immediately after their collision.
\hfill \mbox{\textit{OCR M1 2015 Q6 [14]}}