Answer to Question #151814 in Mechanics | Relativity for Daniel Kim

Question

The water in a river is running due west with a speed of 10m/s. A boy in a boat tries to cross the river by rowing due south at 5m/s. The velocity of the boat relative to the shore is?
A. 5/3m/s, SW
B. 5.83m/s, SW
C. 12.5m/s, SW
D. 11.18m/s, SW

Answer 1

Explanations & Calculations

• The river flows relative to the ground & the boy tries to drive the boat due south (relative to the river) and ultimately the boat sails in a resultant direction.
• Finding the resultant speed & direction can be calculated as follows.
• Consider the velocities as follows according to the standard notation

$\qquad\qquad \begin{aligned} \small V_{RE}&= \,\,\small \leftarrow10ms^{-1}\\ \small V_{BR}&= \,\,\small \downarrow 5ms^{-1}\\ \small V_{BE}&= \small \text{to be found} \end{aligned}$

• By constructing the velocity triangle followed by relative velocity equation (or just constructing it one go if you feel to ), needed quantity could be calculated.

$\qquad\qquad \begin{aligned} \small V_{BE}&= \small V_{BR}+V_{RE}\\ \small V&=\small \downarrow5+\leftarrow10 \end{aligned}$

• And the velocity triangle becomes

• Using Pythagars thoerom to calculate the value of speed,

$\qquad\qquad \begin{aligned} \small V&= \small \sqrt {5^2+10^2 }\\ &= \small 11.18ms^{-1} \end{aligned}$

• Considering the tangent of the needed angle,

$\qquad\qquad \begin{aligned} \small \tan\theta&= \small \frac{10}{5}=2\\ \small \theta &= \small \tan^{-1}2=63.43^0 \end{aligned}$

• Then the velocity is $\bold{\small 11.18ms^{-1} \,\,S\,63.43^0\,W}$