Askiitians Tutor Team
Last Activity: 15 Days ago
To tackle the problem of a swimmer crossing a river with a current, we need to analyze the situation carefully. The swimmer's speed in still water is A m/s, and the river has a width of d meters with a current flowing at B m/s. Let's break down the conditions for the shortest time taken to reach the opposite bank and the expressions for drift.
Finding the Shortest Time to Cross
To minimize the time taken to cross the river, the swimmer must swim at an angle relative to the current. If the swimmer swims directly across the river, the current will push them downstream, increasing the time taken to reach the opposite bank. The key is to find the angle at which the swimmer should swim to counteract the current effectively.
Condition for Shortest Time
The condition for the shortest time involves swimming at an angle θ to the current. The swimmer's effective velocity across the river can be expressed as:
- Vertical component: A * sin(θ)
- Horizontal component: A * cos(θ)
To reach the opposite bank in the shortest time, the vertical component of the swimmer's speed must equal the width of the river divided by the time taken to cross:
Time taken (T) to cross the river is given by:
T = d / (A * sin(θ))
Expression for Shortest Time
To find the shortest time, we need to consider the horizontal drift caused by the current. The drift (D) is given by:
D = B * T = B * (d / (A * sin(θ)))
Thus, the total time taken to cross the river while compensating for the current is:
T = d / (A * sin(θ)) + (B * d) / (A * sin(θ))
To minimize T, we can differentiate this expression with respect to θ and set the derivative to zero, leading to the optimal angle θ that minimizes the time.
Understanding Drift and Its Minimum
Drift refers to the distance the swimmer is carried downstream by the current while crossing the river. The expression for drift can be derived from the time taken to cross and the speed of the current.
Expression for Drift
The drift (D) can be expressed as:
D = B * T = B * (d / (A * sin(θ)))
Minimum Drift Condition
To find the minimum drift, we need to analyze the relationship between the swimmer's speed, the current, and the angle. The minimum drift occurs when the swimmer's angle is optimized to balance the effects of the current. This can be achieved by setting:
tan(θ) = B / A
From this relationship, we can derive the minimum drift as:
D_min = (B * d) / sqrt(A^2 - B^2)
In summary, the swimmer must swim at an angle that balances their speed and the current to minimize the time taken to cross the river while also controlling the drift. By applying these principles, you can effectively analyze similar problems involving motion in a current.
