To solve the problem of two boats moving away from a buoy in a river, we need to analyze their motion based on their velocities and the river's current. Let's break this down step by step.
Understanding the Setup
We have two boats, A and B, moving away from a buoy anchored in the middle of a river. Boat A travels along the river (in the direction of the current), while Boat B moves across the river (perpendicular to the current). Both boats move the same distance away from the buoy and then return.
Defining Variables
Let’s define some variables to make our calculations easier:
- v_s: Velocity of the stream (current).
- v_a: Velocity of Boat A with respect to water, which is n times greater than the stream velocity: v_a = n * v_s.
- v_b: Velocity of Boat B with respect to water, also v_b = n * v_s.
- d: Distance each boat travels away from the buoy.
Calculating Time for Each Boat
Now, let’s calculate the time taken for each boat to travel the distance d and return.
Boat A's Motion
Boat A moves downstream, so its effective velocity is the sum of its speed and the stream velocity:
v_{effective, A} = v_a + v_s = n * v_s + v_s = (n + 1) * v_s.
The time taken for Boat A to travel distance d away from the buoy is:
T(A) = d / v_{effective, A} = d / ((n + 1) * v_s).
Since it has to return the same distance, the total time for Boat A is:
T(A)_{total} = 2 * T(A) = 2 * (d / ((n + 1) * v_s)) = (2d) / ((n + 1) * v_s).
Boat B's Motion
Boat B moves across the river. Its effective velocity is the same as its speed since it is moving perpendicular to the current:
v_{effective, B} = v_b = n * v_s.
The time taken for Boat B to travel distance d away from the buoy is:
T(B) = d / v_{effective, B} = d / (n * v_s).
Similarly, for the return trip, the total time for Boat B is:
T(B)_{total} = 2 * T(B) = 2 * (d / (n * v_s)) = (2d) / (n * v_s).
Finding the Ratio of Times
Now, we can find the ratio of the total times for the two boats:
T(A)_{total} / T(B)_{total} = ((2d) / ((n + 1) * v_s)) / ((2d) / (n * v_s)).
The 2d and v_s cancel out, leading to:
T(A)_{total} / T(B)_{total} = n / (n + 1).
Final Result
Thus, the ratio of the times of motion of the boats is:
T(A) / T(B) = n / (n + 1).
This means that if the velocity of each boat is n times greater than the stream velocity, the time taken by Boat A (moving downstream) to the time taken by Boat B (moving across) is in the ratio of n to (n + 1).
