Standard +0.3 This is a straightforward two-stage projectile problem requiring calculation of range and time of flight using standard SUVAT equations, followed by a simple distance/speed calculation for sound travel. While it involves multiple steps (5-6 marks typical), each step uses routine mechanics formulas with no conceptual difficulty or novel insight required. Slightly above average due to the two-stage nature, but well within standard M1 fare.
5 A small object is projected over horizontal ground from a point O at ground level and makes a loud noise on landing. It has an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(35 ^ { \circ }\) to the horizontal.
Assuming that air resistance on the object may be neglected and that the speed of sound in air is \(343 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate how long after projection the noise is heard at O .
FT for their time Condone consistent \(s \leftrightarrow c\) error (which could lead to correct answer here).
Required time for sound is \(x/343\)
M1
FT from their \(x\)
Total time is \(3.511692\ldots + 0.251598\ldots = 3.76329\ldots\) so \(3.76\) (3 s.f.)
A1
cao following fully correct working throughout question.
8
**Usual notation**
**either** consider height:
Attempt to substitute for $u$ and $a$ in $s = ut + \frac{1}{4}at^2$ | M1 | Accept: $g$ as $g$, $±9.8$, $±9.81$, $±10$; $u = 30$; $s \leftrightarrow c$.
$y = 30sin 35 \cdot t - 4.9t^2$ | A1 | Derivation need not be shown
Need $y = 0$ for time of flight $T$ | B1 |
giving $T = \frac{30sin 35}{4.9}$ $( = 3.511692\ldots)$ | A1 | cao. Any form. May not be explicit.
**Or Consider time to top**
Attempt to substitute for $u$ and $a$ in $v = u + at$ | M1 | Accept: $g$ as $g$, $±9.8$, $±9.81$, $±10$; $u = 30$; $s \leftrightarrow c$.
$v = 30sin 35 - 9.8t$ | A1 | Derivation need not be shown
Need $v = 0$ and to double for time of flight $T$ | B1 |
giving $T = \frac{30sin 35}{4.9}$ $( = 3.511692\ldots)$ | A1 | cao. Any form. May not be explicit.
**then**
$x = 30cos 35 \cdot T$ | M1 | Accept $s \leftrightarrow c$ if consistent with above
so $x = 30cos 35 \times \frac{30sin 35}{4.9}$ $( = 86.29830\ldots)$ | F1 | FT for their time Condone consistent $s \leftrightarrow c$ error (which could lead to correct answer here).
Required time for sound is $x/343$ | M1 | FT from their $x$
Total time is $3.511692\ldots + 0.251598\ldots = 3.76329\ldots$ so $3.76$ (3 s.f.) | A1 | cao following fully correct working throughout question.
| | 8 |
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5 A small object is projected over horizontal ground from a point O at ground level and makes a loud noise on landing. It has an initial speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $35 ^ { \circ }$ to the horizontal.
Assuming that air resistance on the object may be neglected and that the speed of sound in air is $343 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, calculate how long after projection the noise is heard at O .
\hfill \mbox{\textit{OCR MEI M1 2011 Q5 [8]}}