The water in the river moves with a speed of 1.46 km/h. Alice and Bob can swim with a speed of 2.18 km/h with respect to the water. They want to get from point A on one bank of the... The water in the river moves with a speed of 1.46 km/h. Alice and Bob can swim with a speed of 2.18 km/h with respect to the water. They want to get from point A on one bank of the river to point B directly across on the other bank. Alice swims along the straight line AB, while Bob swims with a velocity perpendicular to the river and lets the flow take him some distance downstream. After that, he walks to point B along the river. What is the speed of his walk if he reaches point B at the same time as Alice?
Understand the Problem
The question requires us to compare the swimming techniques of Alice and Bob to find Bob's walking speed such that both reach point B simultaneously. Alice is swimming directly across the river, while Bob swims perpendicular to the flow and uses the river current to wash him downstream before walking to point B. We will calculate the distance each covers and the time taken to solve for Bob's walking speed.
Answer
$$ v_w = v_c \cdot \frac{v_B}{v_A - v_B} $$
Answer for screen readers
The walking speed of Bob is given by: $$ v_w = v_c \cdot \frac{v_B}{v_A - v_B} $$
Steps to Solve
- Identify the distance to be covered by Alice
Assuming the width of the river is ( w ) meters, Alice is swimming directly across the river. Therefore, the distance she covers is simply the width of the river:
- Distance Alice swims: ( d_A = w )
- Establish Alice's speed and time taken
Let Alice's swimming speed be ( v_A ) meters per second. The time ( t_A ) taken by Alice to swim across is: $$ t_A = \frac{d_A}{v_A} = \frac{w}{v_A} $$
- Determine the distance Bob needs to swim
Bob swims at speed ( v_B ) meters per second and must swim at an angle to use the current effectively. He swims perpendicular to the current, meaning that the current pushes him downstream while he swims towards point B. The distance he swims to reach the shore is the same ( w ) meters (the width of the river).
- Calculate Bob's time to swim across the river
Bob's swimming distance is also ( w ), and his time taken, ( t_B ), to swim across is: $$ t_B = \frac{d_B}{v_B} = \frac{w}{v_B} $$
- Calculate the downstream distance caused by the current
The river current has a speed ( v_c ) meters per second. The time Alice takes to swim is equal to the time Bob takes to swim across, meaning Bob can't reach point B until he walks after swimming. The time for Bob includes the time taken to swim and walk. The downstream distance caused by the current in time ( t_B ) is: $$ d_{downstream} = v_c \cdot t_B = v_c \cdot \frac{w}{v_B} $$
- Calculate Bob's walking speed to reach point B
Let ( d_W ) be the distance Bob needs to walk after swimming, which is the downstream distance: $$ d_W = d_{downstream} = v_c \cdot \frac{w}{v_B} $$ If Bob's walking speed is ( v_w ), the time taken to walk ( d_W ) is: $$ t_{walk} = \frac{d_W}{v_w} = \frac{v_c \cdot \frac{w}{v_B}}{v_w} $$
- Set the total time for Bob equal to Alice's time
Both Alice and Bob arrive at point B simultaneously, so we equate their total times: $$ t_A = t_B + t_{walk} $$ This simplifies to: $$ \frac{w}{v_A} = \frac{w}{v_B} + \frac{v_c \cdot \frac{w}{v_B}}{v_w} $$
- Isolate Bob's walking speed (v_w)
To find ( v_w ), we manipulate the equation by isolating ( v_w ): After rearranging, we find: $$ v_w = v_c \cdot \frac{v_B}{v_A - v_B} $$
The walking speed of Bob is given by: $$ v_w = v_c \cdot \frac{v_B}{v_A - v_B} $$
More Information
This formula shows how Bob can adjust his walking speed based on the river's current and their swimming speeds to reach the intended point at the same time as Alice. It's an interesting application of physics and math, illustrating the effect of currents on movement.
Tips
- Failing to correctly equate the times for both swimmers.
- Misunderstanding the effect of the current when calculating Bob's total distance.
- Not properly isolating ( v_w ) in the final formula.
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