Question #108855

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

Expert's answer

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

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

- Additionally its velocity,

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

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