Please can you check this?
A light aeroplane has a cruising speed in still air of \(\displaystyle 40{\rm ms}^{ - 1}\). It is pointed in the direction \(\displaystyle {\rm S21}^ \circ W\), but flies in a wind of speed \(\displaystyle {\rm 18ms}^{{\rm - 1}}\) from the direction \(\displaystyle {\rm N73}^ \circ W\).
Take i to be \(\displaystyle 1{\rm ms}^{ - 1}\) due east and j to be \(\displaystyle 1{\rm ms}^{ - 1}\) due north. Take \(\displaystyle v\) to be the resultant velocity of the aeroplane.
Through previous calculations it has been shown that the resultant velocity \(\displaystyle v\) of the aeroplane is given in component form by
\(\displaystyle v = 2.8788i - 42.6059j\)
a) Find the overall speed \(\displaystyle \left| v \right|\) of the aeroplane (to 4 s.f) and its direction of travel, as a bearing (with the angle to 1 d.p)
b) How long does it take the aeroplane to travel 1 kilometre? How far east does it travel in this time?. Give you answers to 3 s.f.
Here is my working:
a) The overall speed=\(\displaystyle \left| v \right| = \sqrt {2.8722^2 + 42.6059^2 } = 42.70{\rm (4 s}{\rm .f)}\)
The direction of travel \(\displaystyle \approx \arctan \left| {\frac{{42.6059}}{{2.8788}}} \right| \approx 86.1{\rm (1 d}{\rm .p)}\)
This is the 4th quadrant so is -86.1, which corresponds to bearing \(\displaystyle {\rm S 3.9}^ \circ E\)
b) The j component is the rate of progress. It travels 42.6059 ms/s so to travel 1000m takes \(\displaystyle \frac{{{\rm 1000}}}{{{\rm 42}{\rm .6059}}} \approx 23.4709{\rm secs}\)
In that time it would have travelled 23.4709 x 2.8788 = 67.6m east or approx 70.0m (3 s.f)
Please can you check my calculations, especially for part b) and I'm not sure about how to express the significant figures?
Thank you
A light aeroplane has a cruising speed in still air of \(\displaystyle 40{\rm ms}^{ - 1}\). It is pointed in the direction \(\displaystyle {\rm S21}^ \circ W\), but flies in a wind of speed \(\displaystyle {\rm 18ms}^{{\rm - 1}}\) from the direction \(\displaystyle {\rm N73}^ \circ W\).
Take i to be \(\displaystyle 1{\rm ms}^{ - 1}\) due east and j to be \(\displaystyle 1{\rm ms}^{ - 1}\) due north. Take \(\displaystyle v\) to be the resultant velocity of the aeroplane.
Through previous calculations it has been shown that the resultant velocity \(\displaystyle v\) of the aeroplane is given in component form by
\(\displaystyle v = 2.8788i - 42.6059j\)
a) Find the overall speed \(\displaystyle \left| v \right|\) of the aeroplane (to 4 s.f) and its direction of travel, as a bearing (with the angle to 1 d.p)
b) How long does it take the aeroplane to travel 1 kilometre? How far east does it travel in this time?. Give you answers to 3 s.f.
Here is my working:
a) The overall speed=\(\displaystyle \left| v \right| = \sqrt {2.8722^2 + 42.6059^2 } = 42.70{\rm (4 s}{\rm .f)}\)
The direction of travel \(\displaystyle \approx \arctan \left| {\frac{{42.6059}}{{2.8788}}} \right| \approx 86.1{\rm (1 d}{\rm .p)}\)
This is the 4th quadrant so is -86.1, which corresponds to bearing \(\displaystyle {\rm S 3.9}^ \circ E\)
b) The j component is the rate of progress. It travels 42.6059 ms/s so to travel 1000m takes \(\displaystyle \frac{{{\rm 1000}}}{{{\rm 42}{\rm .6059}}} \approx 23.4709{\rm secs}\)
In that time it would have travelled 23.4709 x 2.8788 = 67.6m east or approx 70.0m (3 s.f)
Please can you check my calculations, especially for part b) and I'm not sure about how to express the significant figures?
Thank you