Question

A boat takes 3.0 hours to travel 30 km down a river, then 5.0 hours to return. How fast is the river flowing?

Step by Step with Explanation please. I have trouble getting this relative motion stuff down.

Answer 1

Concepts and reason

First assume the velocity of the river is ${v_r}$ and the velocity of boat is ${v_b}$ . Then, calculate the velocity of down the river and then when boat goes up.

Finally use the relation between speed, time, and distance to calculate the time and solve the equations to find the velocity of the river.

Fundamentals

The relation between the speed, time, and distance is,

$v = \frac{d}{t}$

Here, d is the distance, t is the time, and v is the speed.

The velocity of the river downstream is calculated using the expression as follows;

${v_{{\rm{down stream}}}} = {v_b} + {v_r}$

Here, ${v_b}$ is the velocity of boat, ${v_r}$ is the velocity of river, and ${V_{{\rm{down stream}}}}$ is the velocity downstream.

The velocity of the river upstream is calculated using the expression as follows;

${V_{{\rm{up stream}}}} = {v_b} - {v_r}$

Here, ${v_b}$ is the velocity of boat, ${v_r}$ is the velocity of river, and ${V_{{\rm{up stream}}}}$ is the velocity upstream.

The relation between the speed, time, and distance is,

$v = \frac{d}{t}$

Here, d is the distance, t is the time, and v is the speed.

Substitute 3.0 h for t, 30.0 km for d and ${v_b} + {v_r}$ for v to calculate the downstream velocity of river.

$\begin{array}{c}\\{v_b} + {v_r} = \frac{{30.0\,{\rm{km}}}}{{3.0\,{\rm{h}}}}\\\\ = 10\,{\rm{km/h}}\\\end{array}$

Substitute 5.0 h for t, 30.0 km for d and ${v_b} - {v_r}$ for s to calculate the upstream velocity of river.

$\begin{array}{c}\\{v_b} - {v_r} = \frac{{30.0\,{\rm{km}}}}{{5.0\,{\rm{h}}}}\\\\ = 6\,{\rm{km/h}}\\\end{array}$

Add equation ${v_b} + {v_r} = 10\,{\rm{km/h}}$ with equation ${v_b} - {v_r} = 6\,{\rm{km/h}}$ and solve for ${v_b}.$

$\begin{array}{c}\\{v_b} + {v_r} + {v_b} - {v_r} = 10\,{\rm{km/h}} + 6\,{\rm{km/h}}\\\\2{v_b} = 16\,{\rm{km/h}}\\\\{v_b} = 8\,{\rm{km/h}}\\\end{array}$

Substitute 8 km/h for ${v_b}$ in the above equation ${v_b} + {v_r} = 10\,{\rm{km/h}}$ and solve for ${v_r}.$

$\begin{array}{c}\\8\,{\rm{km/h}} + {v_r} = 10\,{\rm{km/h}}\\\\{v_r} = 2\,{\rm{km/h}}\\\end{array}$

Ans:

The flowing speed of river is 2.0 km/h.