Answer to Question #108855 in Mechanics | Relativity for Cindy

Question

A motor boat is heading due north and crosses a wide river with a speed of 4m/s relative to the water. the water in the river has a uniform speed of 3m/s due east relative to the earth . determine the direction of the motion of the motor boat relative to the earth

Answer 1

Notations

• Velocity of the boat relative to the river: V_(b.r)
• Velocity of the river relative to earth v_(r,e)
• Velocity of the boat relative to earth: V_(b,e)
• Angle between V_(b,e) & V_(r,e) velocity vectors $\theta$

Calculations

• Considering the vector triangle which could be formed by all the velocity vectors the boat is exposed to (Refer the sketch attached),

$\qquad \begin{aligned} \small \end{aligned}$ $\qquad \begin{aligned} \small \tan{\theta}&=\small \frac{V_{b,r}}{V_{r,e}}\\ \small \theta&= \small \tan^{-1}(\frac{4\,ms^{-1}}{3\,{ms^{-1}}})\\ \small &= \small \bold{53.13^{\degree}} \end{aligned}$

Therefor, the direction of the boat is $\small \bold {53.13^{\degree}}$ to north from east.

• Additionally its velocity,

$\qquad \qquad \begin{aligned} \small V^2_{b,e}& = \small V^2_{b,r}\,+\,V^2_{r,e}\\ \small V_{b,e}&=\small \sqrt{(4ms^{-1})^2\,+\,(3ms^{-1})^2}\\ \small&=\small \bold{5ms^{-1}} \end{aligned}$