Vectors, Component Form and Geometric Form

nojcarter · #1 $Old$ January 14th, 2009, 03:07 AM

Hi,

I'm having great difficulty in getting started with this question let alone solving it, Any help would be much appreciated.

A boat that has a speed in still water or 1.8ms^-1 crosses a river that flows due north at a speed of 0.54ms^ -1. It starts from a point on the east bank and is pointed in the direction S33W. Take i to be 1ms^-1 due east and j to be 1ms^-1 due north. Also, take

Vb to be the velocity of the boat in still water,
Vr to be the velocity of the river,
V to be the resultant velocity of the boat.

(a) Express each of the vectors Vb and Vr in component form, giving the components to 4dp.

(b) Hence show that the resultant velocity V of the boat is given in component form, approximately by V = -0.9804i - 0.9696j

(c) By putting V into geometric form, find the overall speed |V| of the boat, (in ms^-1 to 2dp), and its direction of travel as a bearing (with the angel correct to 1dp).

(d) Suppose that the river has a constant width of 24 metres. how long (in secs) does it take the boat to cross the river, and how far (in metres) upstream or downstream has it travelled from its starting point? give answers to 1dp.

Thanks
Jon

earboth · #2 $Old$ January 14th, 2009, 07:34 AM

Quote:

Originally Posted by nojcarter $View Post$

Hi,

I'm having great difficulty in getting started with this question let alone solving it, Any help would be much appreciated.

A boat that has a speed in still water or 1.8ms^-1 crosses a river that flows due north at a speed of 0.54ms^ -1. It starts from a point on the east bank and is pointed in the direction S33W. Take i to be 1ms^-1 due east and j to be 1ms^-1 due north. Also, take

Vb to be the velocity of the boat in still water,
Vr to be the velocity of the river,
V to be the resultant velocity of the boat.

(a) Express each of the vectors Vb and Vr in component form, giving the components to 4dp.

(b) Hence show that the resultant velocity V of the boat is given in component form, approximately by V = -0.9804i - 0.9696j

(c) By putting V into geometric form, find the overall speed |V| of the boat, (in ms^-1 to 2dp), and its direction of travel as a bearing (with the angel correct to 1dp).

(d) Suppose that the river has a constant width of 24 metres. how long (in secs) does it take the boat to cross the river, and how far (in metres) upstream or downstream has it travelled from its starting point? give answers to 1dp.

Thanks
Jon

1. The positive x-axis points East.

2. The angle of S33°W equals an angle of 237° measured counterclockwise from the positive x-axis.

3. Using this angle the vectors $\overrightarrow{v_b}$ and $\overrightarrow{v_r}$ can be written as:

$\overrightarrow{v_b} = 1.8 \cdot (\cos(237^\circ),\ \sin(237^\circ)) = (-0.98035,\ -1.50961)$

$\overrightarrow{v_r} = 0.54 \cdot (\cos(90^\circ),\ \sin(90^\circ)) = (0, 0.54)$

4. $\vec v = \overrightarrow{v_b} + \overrightarrow{v_r}$

5. Calculate the length of $\vec v$ to get the speed of the boat.

6. To calculate the time which it takes to cross the river you only have to use the horizontal component of $\overrightarrow{v_b}$ .

nojcarter · #3 $Old$ January 15th, 2009, 02:41 AM

Hi earboth,

Thanks for your help, I've managed to wriggle my way through the question and have submitted it just in time.

Thanks
Jon