Solution to Test Paper 3
From Mechanics
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| 1 (a) | | 1 (a) | ||
|<math>\begin{align} | |<math>\begin{align} | ||
| - | & 0=24.5-9.8t \\ | + | & 0=24\textrm{.}5-9\textrm{.}8t \\ |
| - | & t=\frac{24.5}{9.8}=2.5\text{ seconds} \\ | + | & t=\frac{24\textrm{.}5}{9\textrm{.}8}=2\textrm{.}5\text{ seconds} \\ |
\end{align}</math> | \end{align}</math> | ||
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| 1 (b) | | 1 (b) | ||
| <math>\begin{align} | | <math>\begin{align} | ||
| - | & {{0}^{2}}={{24.5}^{2}}+2\times (-9.8)\times s \\ | + | & {{0}^{2}}={{24\textrm{.}5}^{2}}+2\times (-9\textrm{.}8)\times s \\ |
| - | & s=\frac{{{24.5}^{2}}}{2\times 9.8}=30.625\text{ m} \\ | + | & s=\frac{{{24\textrm{.}5}^{2}}}{2\times 9\textrm{.}8}=30\textrm{.}625\text{ m} \\ |
| + | & \\ | ||
| + | & \text{OR} \\ | ||
| + | & \\ | ||
| + | & s=\frac{1}{2}(24\textrm{.}5+0)\times 2\textrm{.}5=30\textrm{.}625\text{ m} \\ | ||
\end{align}</math> | \end{align}</math> | ||
| - | |||
| - | OR | ||
| - | |||
| - | <p> | ||
| - | |||
| - | <math>s=\frac{1}{2}(24.5+0)\times 2.5=30.625\text{ m}</math> | ||
| - | |||
| - | </p> | ||
| (3 marks) | | (3 marks) | ||
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| 2 (c) | | 2 (c) | ||
| <math>\begin{align} | | <math>\begin{align} | ||
| - | & T-4900=500\times 0.6 \\ | + | & T-4900=500\times 0\textrm{.}6 \\ |
& T=300+4900=5200\text{ N} \\ | & T=300+4900=5200\text{ N} \\ | ||
\end{align}</math> | \end{align}</math> | ||
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|- | |- | ||
| - | | | + | | 2 (d) |
| <math>\begin{align} | | <math>\begin{align} | ||
& 4800-4900=500a \\ | & 4800-4900=500a \\ | ||
| - | & a=\frac{-100}{500}=-0.2\text{ m}{{\text{s}}^{\text{-2}}} \\ | + | & a=\frac{-100}{500}=-0\textrm{.}2\text{ m}{{\text{s}}^{\text{-2}}} \\ |
| - | & 0.\text{2 m}{{\text{s}}^{-\text{2}}}\quad \text{downwards} \\ | + | & 0\textrm{.}\text{2 m}{{\text{s}}^{-\text{2}}}\quad \text{downwards} \\ |
\end{align}</math> | \end{align}</math> | ||
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|- | |- | ||
| 3 (a) | | 3 (a) | ||
| - | |<math>R=5\times 9.8\cos 42{}^\circ =36.4\text{ N}</math> | + | |<math>R=5\times 9\textrm{.}8\cos 42{}^\circ =36\textrm{.}4\text{ N}</math> |
| (2 mark) | | (2 mark) | ||
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|- | |- | ||
| 3 (b) | | 3 (b) | ||
| - | | <math>F=0.4R=14.6\text{ N}</math> | + | | <math>F=0\textrm{.}4R=14\textrm{.}6\text{ N}</math> |
| (2 marks) | | (2 marks) | ||
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| 3 (c) | | 3 (c) | ||
| <math>\begin{align} | | <math>\begin{align} | ||
| - | & 5\times 9.8\sin 42{}^\circ -14.6=5a \\ | + | & 5\times 9\textrm{.}8\sin 42{}^\circ -14\textrm{.}6=5a \\ |
| - | & a=\frac{18.22}{5}=3.64\text{ m}{{\text{s}}^{\text{-2}}} \\ | + | & a=\frac{18\textrm{.}22}{5}=3\textrm{.}64\text{ m}{{\text{s}}^{\text{-2}}} \\ |
\end{align}</math> | \end{align}</math> | ||
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| 3 (d) | | 3 (d) | ||
| <math>\begin{align} | | <math>\begin{align} | ||
| - | & 2=\frac{1}{2}\times 3.64{{t}^{2}} \\ | + | & 2=\frac{1}{2}\times 3\textrm{.}64{{t}^{2}} \\ |
| - | & t=\sqrt{\frac{4}{3.64}}=1.05\text{ seconds} \\ | + | & t=\sqrt{\frac{4}{3\textrm{.}64}}=1\textrm{.}05\text{ seconds} \\ |
\end{align}</math> | \end{align}</math> | ||
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| 4 (a) | | 4 (a) | ||
| <math>\begin{align} | | <math>\begin{align} | ||
| - | & Q\cos 40{}^\circ =8\times 9.8\sin 40{}^\circ \\ | + | & Q\cos 40{}^\circ =8\times 9\textrm{.}8\sin 40{}^\circ \\ |
| - | & Q=\frac{8\times 9.8\sin 40{}^\circ }{\cos 40{}^\circ }=65.8\text{ N} \\ | + | & Q=\frac{8\times 9\textrm{.}8\sin 40{}^\circ }{\cos 40{}^\circ }=65\textrm{.}8\text{ N} \\ |
\end{align}</math> | \end{align}</math> | ||
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| 4 (b) | | 4 (b) | ||
|<math>\begin{align} | |<math>\begin{align} | ||
| - | & Q\cos 40{}^\circ +F=8\times 9.8\sin 40{}^\circ \\ | + | & Q\cos 40{}^\circ +F=8\times 9\textrm{.}8\sin 40{}^\circ \\ |
| - | & Q\sin 40{}^\circ +8\times 9.8\cos 40{}^\circ =R \\ | + | & Q\sin 40{}^\circ +8\times 9\textrm{.}8\cos 40{}^\circ =R \\ |
| - | & Q\cos 40{}^\circ =8\times 9.8\sin 40{}^\circ \\ | + | & Q\cos 40{}^\circ =8\times 9\textrm{.}8\sin 40{}^\circ \\ |
| - | & \text{ }-0.3(Q\sin 40{}^\circ +8\times 9.8\cos 40{}^\circ ) \\ | + | & \text{ }-0\textrm{.}3(Q\sin 40{}^\circ +8\times 9\textrm{.}8\cos 40{}^\circ ) \\ |
| - | & Q=\frac{8\times 9.8\sin 40{}^\circ -0. | + | & Q=\frac{8\times 9\textrm{.}8\sin 40{}^\circ -0.\textrm{.}\times 8\times 9\textrm{.}8\cos 40{}^\circ }{\cos 40{}^\circ +0.\times \sin 40{}^\circ } \\ |
| - | & \text{ }=33.8\text{ N} \\ | + | & \text{ }=33\textrm{.}8\text{ N} \\ |
\end{align}</math> | \end{align}</math> | ||
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|- | |- | ||
| 5 (a) | | 5 (a) | ||
| - | | <math>\mathbf{v}=(\mathbf{i}+\mathbf{j})+(0.05\mathbf{i}-0.1\mathbf{j})t</math> | + | | <math>\mathbf{v}=(\mathbf{i}+\mathbf{j})+(0\textrm{.}05\mathbf{i}-0\textrm{.}1\mathbf{j})t</math> |
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| 5 (b) | | 5 (b) | ||
| <math>\begin{align} | | <math>\begin{align} | ||
| - | & -(1+0.05t)=1-0.1t \\ | + | & -(1+0\textrm{.}05t)=1-0\textrm{.}1t \\ |
| - | & 0.05t=2 \\ | + | & 0\textrm{.}05t=2 \\ |
& t=40 \\ | & t=40 \\ | ||
\end{align}</math> | \end{align}</math> | ||
Current revision
| 1 (a) | \displaystyle \begin{align}
& 0=24\textrm{.}5-9\textrm{.}8t \\ & t=\frac{24\textrm{.}5}{9\textrm{.}8}=2\textrm{.}5\text{ seconds} \\ \end{align} | (3 marks)
|
| 1 (b) | \displaystyle \begin{align}
& {{0}^{2}}={{24\textrm{.}5}^{2}}+2\times (-9\textrm{.}8)\times s \\ & s=\frac{{{24\textrm{.}5}^{2}}}{2\times 9\textrm{.}8}=30\textrm{.}625\text{ m} \\ & \\ & \text{OR} \\ & \\ & s=\frac{1}{2}(24\textrm{.}5+0)\times 2\textrm{.}5=30\textrm{.}625\text{ m} \\ \end{align} | (3 marks) |
| 1 (c) | 61.25 m
| (1 marks) |
| (7 marks)
| ||
| 2 (a) | 
| (1 marks)
|
| 2 (b) | T = 4900 N | (1 marks)
|
| 2 (c) | \displaystyle \begin{align}
& T-4900=500\times 0\textrm{.}6 \\ & T=300+4900=5200\text{ N} \\ \end{align} | (3 mark) |
| 2 (d) | \displaystyle \begin{align}
& 4800-4900=500a \\ & a=\frac{-100}{500}=-0\textrm{.}2\text{ m}{{\text{s}}^{\text{-2}}} \\ & 0\textrm{.}\text{2 m}{{\text{s}}^{-\text{2}}}\quad \text{downwards} \\ \end{align} | (4 mark) |
| (9 marks)
| ||
| 3 (a) | \displaystyle R=5\times 9\textrm{.}8\cos 42{}^\circ =36\textrm{.}4\text{ N} | (2 mark)
|
| 3 (b) | \displaystyle F=0\textrm{.}4R=14\textrm{.}6\text{ N} | (2 marks)
|
| 3 (c) | \displaystyle \begin{align}
& 5\times 9\textrm{.}8\sin 42{}^\circ -14\textrm{.}6=5a \\ & a=\frac{18\textrm{.}22}{5}=3\textrm{.}64\text{ m}{{\text{s}}^{\text{-2}}} \\ \end{align} | (3 marks)
|
| 3 (d) | \displaystyle \begin{align}
& 2=\frac{1}{2}\times 3\textrm{.}64{{t}^{2}} \\ & t=\sqrt{\frac{4}{3\textrm{.}64}}=1\textrm{.}05\text{ seconds} \\ \end{align}
| (3 marks)
|
| (10 marks)
| ||
| 4 (a) | \displaystyle \begin{align}
& Q\cos 40{}^\circ =8\times 9\textrm{.}8\sin 40{}^\circ \\ & Q=\frac{8\times 9\textrm{.}8\sin 40{}^\circ }{\cos 40{}^\circ }=65\textrm{.}8\text{ N} \\ \end{align} | (3 mark)
|
| 4 (b) | \displaystyle \begin{align}
& Q\cos 40{}^\circ +F=8\times 9\textrm{.}8\sin 40{}^\circ \\ & Q\sin 40{}^\circ +8\times 9\textrm{.}8\cos 40{}^\circ =R \\ & Q\cos 40{}^\circ =8\times 9\textrm{.}8\sin 40{}^\circ \\ & \text{ }-0\textrm{.}3(Q\sin 40{}^\circ +8\times 9\textrm{.}8\cos 40{}^\circ ) \\ & Q=\frac{8\times 9\textrm{.}8\sin 40{}^\circ -0.\textrm{.}\times 8\times 9\textrm{.}8\cos 40{}^\circ }{\cos 40{}^\circ +0.\times \sin 40{}^\circ } \\ & \text{ }=33\textrm{.}8\text{ N} \\ \end{align} | (5 marks)
|
| (8 marks)
| ||
| 5 (a) | \displaystyle \mathbf{v}=(\mathbf{i}+\mathbf{j})+(0\textrm{.}05\mathbf{i}-0\textrm{.}1\mathbf{j})t
| (2 marks)
|
| 5 (b) | \displaystyle \begin{align}
& -(1+0\textrm{.}05t)=1-0\textrm{.}1t \\ & 0\textrm{.}05t=2 \\ & t=40 \\ \end{align} | (4 marks)
|
| (6 marks)
|
