Question

A boat travels at a speed of \(v_{\mathrm{BW}}\) relative to the water in a river of width \(D .\) The speed at which the water is flowing is \(v_{\mathrm{W}}\) a) Prove that the time required to cross the river to a point exactly opposite the starting point and then to return is \(T_{1}=2 D / \sqrt{v_{B W}^{2}-v_{W}^{2}}\) b) Also prove that the time for the boat to travel a distance \(D\) downstream and then return is \(T_{1}=2 D v_{\mathrm{B}} /\left(v_{\mathrm{BW}}^{2}-v_{\mathrm{w}}^{2}\right)\)

Short answer

Question: Calculate the time it takes for a boat to cross a river of width D and return to its starting point in two different scenarios: a) The boat aims to reach exactly the opposite point on the shore, and b) the boat travels a distance D downstream and then returns to the starting point. The boat's velocity relative to water is \(v_{\mathrm{BW}}\) and the river's current velocity is \(v_{\mathrm{W}}\). Answer: a) \(T_1 = 2\frac{D}{\sqrt{v_{\mathrm{BW}}^2 - v_{\mathrm{W}}^2}}\), b) \(T_2 = \frac{2Dv_{\mathrm{BW}}}{v_{\mathrm{BW}}^2 - v_{\mathrm{W}}^2}\).

Step-by-step solution

Determine the boat's actual velocity component perpendicular to the river#

First, let's determine the boat's velocity component perpendicular to the river, which we can call \(v_{\mathrm{B}}\). Since the boat has to overcome the river's current, we can determine \(v_{\mathrm{B}}\) using the Pythagorean theorem: \(v_{\mathrm{B}}^2 = v_{\mathrm{BW}}^2 - v_{\mathrm{W}}^2\). Here, \(v_{BW}\) is the boat's velocity relative to water, and \(v_W\) is the river's current velocity.

Calculate the time required to cross the river#

Now that we have the boat's velocity perpendicular to the river (\(v_{\mathrm{B}}\)), we can calculate the time it takes for the boat to cross the river using the formula \(t = \frac{d}{v}\). Since the river's width is given as \(D\), the time required to cross the river is \(t_{\mathrm{cross}} = \frac{D}{v_{\mathrm{B}}} = \frac{D}{\sqrt{v_{\mathrm{BW}}^2 - v_{\mathrm{W}}^2}}\).

Double the time to account for the return trip#

As the boat turns back and returns exactly to the starting point, the time required for the return trip is the same as the time it took to cross the river initially. Therefore, the total time required (\(T_1\)) is twice the time calculated in Step 2: \(T_1 = 2t_{\mathrm{cross}} = 2 \frac{D}{\sqrt{v_{\mathrm{BW}}^2 - v_{\mathrm{W}}^2}}\). b) Time required to travel a distance \(D\) downstream and return to the starting point

Determine the boat's velocity components downstream and upstream#

We will first find the boat's velocity components downstream (\(v_{\mathrm{DS}}\)) and upstream (\(v_{\mathrm{US}}\)). When the boat is moving downstream, we will add the river's current velocity to the boat's speed relative to water: \(v_{\mathrm{DS}} = v_{\mathrm{BW}} + v_{\mathrm{W}}\). Similarly, when going upstream, we will subtract the river's current velocity from the boat's speed relative to water: \(v_{\mathrm{US}} = v_{\mathrm{BW}} - v_{\mathrm{W}}\).

Calculate the time required to travel downstream and upstream#

We can use the formula \(t = \frac{d}{v}\) again to calculate the time taken when traveling downstream (\(t_{\mathrm{DS}}\)) and upstream (\(t_{\mathrm{US}}\)). Since both distances are given as \(D\), we have \(t_{\mathrm{DS}} = \frac{D}{v_{\mathrm{DS}}} = \frac{D}{v_{\mathrm{BW}} + v_{\mathrm{W}}}\) and \(t_{\mathrm{US}} = \frac{D}{v_{\mathrm{US}}} = \frac{D}{v_{\mathrm{BW}} - v_{\mathrm{W}}}\).

Calculate the total time required for both parts of the trip#

Finally, we will sum up the time taken for both parts of the trip to get the total time \(T_2 = t_{\mathrm{DS}} + t_{\mathrm{US}} = \frac{D}{v_{\mathrm{BW}} + v_{\mathrm{W}}} + \frac{D}{v_{\mathrm{BW}} - v_{\mathrm{W}}}\). This can be simplified by finding the common denominator and combining the fractions: \(T_2 = \frac{2Dv_{\mathrm{BW}}}{v_{\mathrm{BW}}^2 - v_{\mathrm{W}}^2}\).

Key concepts

Boat and River Problems

In physics, particularly when studying relative velocity, one common scenario is the boat and river problem. This situation involves a boat traveling across a river with a flowing current. Understanding the velocities involved is crucial to solving such problems.

• The boat's velocity (\(v_{\mathrm{BW}}\)) is relative to the water.
• The river has its own current velocity (\(v_{\mathrm{W}}\)).
• Key calculations often include crossing the river perpendicularly and returning, or moving downstream and then upstream.

These problems involve analyzing how these velocities interact to determine path and time taken by the boat.

Perpendicular Velocity Component

To solve a boat crossing a river perpendicularly, it's essential to understand the boat's perpendicular velocity component. This component is what allows the boat to travel straight across the river.
When a boat moves across a river, it must overcome the lateral flow of the river's current. The perpendicular velocity is calculated using:

• The Pythagorean theorem: \(v_{\mathrm{B}} = \sqrt{v_{\mathrm{BW}}^{2} - v_{\mathrm{W}}^{2}}\).
• Here, \(v_{\mathrm{B}}\) is the boat's velocity against the current, and it allows for calculating crossing time with classic formulas.

By understanding this velocity component, you can predict and calculate the exact path the boat must follow to reach the opposite point directly across the river.

Downstream and Upstream Motion

When a boat moves downstream, it is aided by the river's current. Conversely, when moving upstream, it must battle against the current. This leads to different effective velocities:

• Downstream velocity: \(v_{\mathrm{DS}} = v_{\mathrm{BW}} + v_{\mathrm{W}}\)
• Upstream velocity: \(v_{\mathrm{US}} = v_{\mathrm{BW}} - v_{\mathrm{W}}\)
• Downstream motion is faster since current adds to the boat's speed.
• Upstream motion is slower as the current subtracts from the boat's speed.

Learning to calculate these velocities helps determine the time for a complete cycle of downstream and upstream travel. Each velocity requires different time calculations, which we will explore next.

Time Calculations in Physics

Physics often requires calculating time for various motions. In boat and river problems, the task involves determining how long it takes for a boat to cross a river or travel a set distance with and against the current.
For time calculations, remember:

• The basic formula is: \(t = \frac{d}{v}\)
• For crossing the river perpendicularly, the time is \(t_{\mathrm{cross}} = \frac{D}{\sqrt{v_{\mathrm{BW}}^{2} - v_{\mathrm{W}}^{2}}}\).
• For a downstream trip, it's \(t_{\mathrm{DS}} = \frac{D}{v_{\mathrm{BW}} + v_{\mathrm{W}}}\).
• And for upstream, it's \(t_{\mathrm{US}} = \frac{D}{v_{\mathrm{BW}} - v_{\mathrm{W}}}\).

These time calculations ensure a comprehensive understanding of the total time (\(T\)) required for various boat navigations, providing a significant edge in physics problem-solving.