| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | SUVAT in 2D & Gravity |
| Type | Particle motion: 2D constant acceleration |
| Difficulty | Moderate -0.8 This is a straightforward application of SUVAT equations in 2D vector form. Part (i) requires direct integration/use of standard formulae (v = u + at, r = ut + ½at²), and part (ii) involves simple trigonometry to convert a velocity vector to a bearing. No problem-solving insight needed, just routine application of mechanics formulae with vectors. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10h Vectors in kinematics: uniform acceleration in vector form3.02e Two-dimensional constant acceleration: with vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Use of \(v = u + ta\) | M1 | substitution required. Must be vectors. |
| \(v = 4i - 2tj\) | A1 | |
| Use of \(r = (r_0 + ut) + \frac{1}{2}ta\) | M1 | substitution required. \(r_0\) not required. Must be vectors. |
| \(+ 3j\) | B1 | May be seen on either side of a meaningful equation for \(r\) |
| \(r = 4ti + (3 - t^2)j\) | A1 | Accept \(r = 3j + 4ti - \frac{1}{2} \times 2 \times t^2 j\) oe written in a correct notation. Isw, providing not reduced to scalar: (see 12c in marking instructions) |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| \(v = \int adt\) | M1 | Attempt at integration. Condone no '+c'. Must be vectors. |
| \(v = 4i - 2tj\) | A1 | cao |
| \(r = \int vdt\) | M1 | Integrate their \(v\) but must contain 2 components. Must be vectors. |
| \(+ 3j\) | B1 | May be seen on either side of a meaningful equation for \(r\) |
| \(r = 4ti + (3 - t^2)j\) | A1 | Accept \(r = 3j + 4ti - \frac{1}{2} \times 2 \times t^2 j\) oe written in a correct notation. Isw, providing not reduced to scalar: (see 12c in marking instructions) |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| \(v(2.5) = 4i - 5j\) | B1 | FT their \(v\) |
| Angle is \((90+)\arctan\frac{5}{4}\) | M1 | Award for arctan attempted oe. FT their values. Allow argument to be \(\pm\) (their i cpt)/(their j cpt) or \(\pm\) (their j cpt)/(their i cpt). Allow this mark if bearing of position vector attempted. |
| \(= 141.34019\ldots\) so \(141°\) (3 s.f.) | A1 | cao |
| 3 | ||
| 8 |
**(i) Either** using suvat:
Use of $v = u + ta$ | M1 | substitution required. Must be vectors.
$v = 4i - 2tj$ | A1 |
Use of $r = (r_0 + ut) + \frac{1}{2}ta$ | M1 | substitution required. $r_0$ not required. Must be vectors.
$+ 3j$ | B1 | May be seen on either side of a meaningful equation for $r$
$r = 4ti + (3 - t^2)j$ | A1 | Accept $r = 3j + 4ti - \frac{1}{2} \times 2 \times t^2 j$ oe written in a correct notation. Isw, providing not reduced to scalar: (see 12c in marking instructions)
| | 5 |
**Or using integration:**
$v = \int adt$ | M1 | Attempt at integration. Condone no '+c'. Must be vectors.
$v = 4i - 2tj$ | A1 | cao
$r = \int vdt$ | M1 | Integrate their $v$ but must contain 2 components. Must be vectors.
$+ 3j$ | B1 | May be seen on either side of a meaningful equation for $r$
$r = 4ti + (3 - t^2)j$ | A1 | Accept $r = 3j + 4ti - \frac{1}{2} \times 2 \times t^2 j$ oe written in a correct notation. Isw, providing not reduced to scalar: (see 12c in marking instructions)
| | 5 |
**(ii)**
$v(2.5) = 4i - 5j$ | B1 | FT their $v$
Angle is $(90+)\arctan\frac{5}{4}$ | M1 | Award for arctan attempted oe. FT their values. Allow argument to be $\pm$ (their i cpt)/(their j cpt) or $\pm$ (their j cpt)/(their i cpt). Allow this mark if bearing of position vector attempted.
$= 141.34019\ldots$ so $141°$ (3 s.f.) | A1 | cao
| | 3 |
| | 8 |
---6 In this question, $\mathbf { i }$ and $\mathbf { j }$ are unit vectors east and north respectively. Position vectors are with respect to an origin O . Time $t$ is in seconds.
A skater has a constant acceleration of $- 2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At $t = 0$, his velocity is $4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and his position vector is $3 \mathbf { j } \mathrm {~m}$.\\
(i) Find expressions in terms of $t$ for the velocity and the position vector of the skater at time $t$.\\
(ii) Calculate as a bearing the direction of motion of the skater when $t = 2.5$.
\hfill \mbox{\textit{OCR MEI M1 2011 Q6 [8]}}