Lesson Wind and Current problems

Wind and Current problems

In this lesson some typical problems of the type "Wind and Current" are presented for a motorboat and airplane making round trips.

In problem 1 and 2 the length of the trip is given, as well as spent time moving in each direction.
A motorboat (airplane) speed in still water (still air) and the current (wind) speed are unknown.
The way to solve these problems is to reduce them to the system of two linear equations with two unknowns, and then to solve this system.

In problem 3 and 4 the duration of the trip is given to each direction, as well as the one of two speeds.
The other speed and the travel length are unknown.
The way to solve these problems is to reduce them to one linear equation with one unknown variable - speed, and then to solve this equation.
When it is done, you can calculate the length of the trip.

Problem 1. Motorboat moving upstream and downstream on a river

A motorboat makes the 24 miles upstream trip on a river against the current in 3 hours.
Returning trip with the current takes 2 hours.
Find the motorboat speed in still water and the current speed.

Solution
Let us denote the motorboat speed in still water as miles per hour, and the current speed as miles per hour.
Then the speed of the motorboat is when it moves upstream and when it is moves downstream.
For the upstream trip we have the equation connecting the speed, the time and the distance in the form
.
For the downstream trip we have similar equation in the form
.
Thus, we have the system of two linear equations with two unknowns
.
Let's open brackets first:
.
Let's apply an elimination method to solve this system of equation.
Multiply both sides of the first equation by 2 and both sides of the second equation by 3:
.
Add the first and the second equations to eliminate variable . You get
.
Divide both side of the last equation by 12. You get
.
Now, substitute this value of to the very first equation. You get the equation with the only one unknown
.
Solve it for making step by step transformations:
,
,
,
.

Answer. The motorboat speed in still water is equal to 10 miles per hour.
The current speed is 2 miles per hour.

Problem 2. Airplane flying into the wind and with the wind

When an airplane flies into the wind, it can travel 3000 miles in 6 hours.
When it flies with the wind, it can travel the same distance in 5 hours.
Find the speed of the airplane in still air and the speed of the wind.

Solution
Let us denote the airplane speed in still air as miles per hour, and the speed of the wind as miles per hour.
Then the speed of the airplane is when it moves into the wind and when it moves with the wind.
For the trip into the wind we have the equation connecting the speed, the time and the distance in the form
.
For the trip with the wind we have similar equation in the form
.
Thus, we have the system of two linear equations with two unknowns
.
Let's open brackets first:
.
Let's apply an elimination method to solve this system of equation.
Multiply both sides of the first equation by 5 and both sides of the second equation by 6:
.
Add the first and the second equations to eliminate variable . You get
.
Divide both side of the last equation by 60. You get
.
Now, substitute this value of to the very first equation. You get the equation with the only one unknown
.
Solve it for making step by step transformations:
,
,
,
.

Answer. The speed of the airplane in still air is equal to 550 miles per hour.
The speed of the wind is 50 miles per hour.

Problem 3. Motorboat moving upstream and downstream on a river

A motorboat makes an upstream trip on a river in 3 hours against the current, which is of 2 miles per hour.
The return downstream trip with the same current takes 2 hours.
Find the motorboat speed in still water and the trip length.

Solution
Let us denote the motorboat speed in still water as miles per hour.
Then the speed of the motorboat is when it moves upstream and when it moves downstream.
The length of the upstream trip is equal to miles.
The length of the downstream trip is equal to miles.
Since this is the same length, this gives us the equation with one unknown
.
Let's open brackets, collect variable terms on the left side, constant terms on the right side and reduce like terms, step by step:
,
,
.
Thus, we just found the motorboat speed in still water as 10 miles per hour.
Now, determine the trip length by substituting into the formula
L = miles.

Answer. The motorboat speed in still water is equal to 10 miles per hour.
The trip length is equal to 24 miles.

Problem 4. Airplane flying into the wind and with the wind

Airplane flies for 6 hours against the wind.
The return fly with the same tail wind takes 5 hours.
The airplane speed in the still air is 550 miles per hour.
Find the wind speed and the fly length.

Solution
Let us denote the wind speed as miles per hour.
Then the speed of the airplane is when it moves against the wind and when it moves with the wind.
The airplane travels for miles when flies against the wind.
The airplane travels for miles when flies with the wind.
Since this is the same length, this gives us the equation with one unknown
.
Let's open brackets, collect variable terms on the left side, constant terms on the right side and reduce like terms, step by step:
,
,
,
.
Thus, we just found the wind speed as 50 miles per hour.
Now, determine the trip length by substituting into the formula
L = miles.

Answer. The wind speed is equal to 50 miles per hour.
The fly length is equal to 3000 miles.