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Chapter 3: Problem 46

A river \(400 \mathrm{~m}\) wide is flowing at a rate of \(2 \cdot 0 \mathrm{~m} / \mathrm{s}\). \(\mathrm{A}\) boat is sailing at a velocity of \(10 \mathrm{~m} / \mathrm{s}\) with respect to the water, in a direction perpendicular to the river. (a) Find the time taken by the boat to reach the opposite bank. (b) How far from the point directly opposite to the starting point does the boat reach the opposite bank ?

Short Answer

Expert verified
(a) 40 seconds; (b) 80 meters from the starting point.

Step by step solution

01

Understanding the Problem

The boat is moving across a river. We need to calculate how long it takes for the boat to cross the river and how far it will drift downstream due to the river's current.
02

Identify Known Values

The width of the river is given as 400 meters, and the boat's velocity relative to the water is 10 meters per second, perpendicular to the river. The current flows at 2 meters per second parallel to the river.
03

Calculate Time to Cross the River

The time taken to cross the river depends only on the boat's velocity perpendicular to the river. Since the width of the river is 400 meters and the boat's perpendicular velocity is 10 m/s, the time taken, \( t \), is calculated as follows: \[ t = \frac{\text{Width of the river}}{\text{Boat's velocity perpendicular to the river}} = \frac{400}{10} = 40\, \text{seconds} \].
04

Calculate Downstream Drift

While the boat crosses the river, it will also drift downstream due to the current. The distance drifted, \( d \), is calculated using the formula: \( d = \text{Current velocity} \times \text{Time} \). Here, \( d = 2 \times 40 = 80 \) meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Relative Velocity
When dealing with moving objects, like a boat crossing a river, understanding relative velocity is crucial. The concept of relative velocity involves analyzing the motion of one object concerning another.
  • In our example, the boat moves relative to the water. This means we measure its speed from the perspective of the moving river water.
  • The boat has a speed of 10 m/s with respect to the water, ignoring any additional motion caused by the current.
The river itself flows at 2 m/s. To fully describe the boat's motion relative to the riverbank, we must account for both movements. By understanding relative velocity, you can better predict where an object will end up based on its speed and the speed of the medium it moves through.
Exploring Vector Components
Understanding vector components is essential when breaking down motion in different directions. In our river crossing problem, motion occurs along two main axes: across the river and down the river.
  • Perpendicular Component: This is the motion across the river. The boat's velocity of 10 m/s represents the perpendicular component as it's directed straight across from bank to bank.
  • Parallel Component: This is the motion along the river. The river's flow velocity of 2 m/s is the parallel component that causes the boat to drift downstream as it crosses.
By separating these components, you can focus on one direction at a time, simplifying calculations and predictions about total displacement and position.
The Role of Perpendicular Motion
When the problem states the boat moves perpendicular to the river, it means the boat's direction is straight across without considering the river flow. Perpendicular motion is essential because it directly affects crossing time.
  • To determine how quickly the boat reaches the opposite bank, only its perpendicular speed (10 m/s) is relevant.
  • The width of the river is 400 meters. To find the time to cross, you divide the width by the boat's perpendicular velocity, giving \[ t = \frac{400}{10} = 40 \text{ seconds} \].
This calculation showcases how straightforward determining crossing time can be when focusing solely on perpendicular motion. Other factors, like drift, are considered separately.
Calculating Boat Drift
Drift calculation determines how far downstream the boat ends up due to the river's current, separate from its perpendicular crossing speed. This calculation is crucial to understanding where the boat lands relative to its starting point.
  • The river current moves parallel to the riverbank at 2 m/s.
  • The time taken to cross, found using perpendicular motion, is 40 seconds.
  • During this time, the boat will drift downstream for \[ d = 2 \times 40 = 80 \text{ meters} \].
Combining drift and perpendicular motion answers the problem fully by explaining both the crossing time and ending position, which is crucial for river crossing problems.

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Most popular questions from this chapter

It is \(260 \mathrm{~km}\) from Patna to Ranchi by air and \(320 \mathrm{~km}\) by road. An aeroplane takes 30 minutes to go from Patna to Ranchi whereas a delux bus takes 8 hours. (a) Find the average speed of the plane. (b) Find the average speed of the bus. (c) Find the average velocity of the plane. (d) Find the average velocity of the bus.

A car travelling at \(60 \mathrm{~km} / \mathrm{h}\) overtakes another car travelling at \(42 \mathrm{~km} / \mathrm{h}\). Assuming each car to be \(5 \cdot 0 \mathrm{~m}\) long, find the time taken during the overtake and the total road distance used for the overtake.

A particle starting from rest moves with constant acceleration. If it takes \(5 \cdot 0 \mathrm{~s}\) to reach the speed \(18 \cdot 0\) \(\mathrm{km} / \mathrm{h}\) find (a) the average velocity during this period, and (b) the distance travelled by the particle during this period.

An aeroplane has to go from a point \(A\) to another point \(B, 500 \mathrm{~km}\) away due \(30^{\circ}\) east of north. A wind is blowing due north at a speed of \(20 \mathrm{~m} / \mathrm{s}\). The air-speed of the plane is \(150 \mathrm{~m} / \mathrm{s}\). (a) Find the direction in which the pilot should head the plane to reach the point \(B .\) (b) Find the time taken by the plane to go from \(A\) to \(B\).

A ball is thrown horizontally from a point \(100 \mathrm{~m}\) above the ground with a speed of \(20 \mathrm{~m} / \mathrm{s}\). Find (a) the time it takes to reach the ground, (b) the horizontal distance it travels before reaching the ground, (c) the velocity (direction and magnitude) with which it strikes the ground.
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