/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 A rowboat crosses a river with a... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 39

A rowboat crosses a river with a velocity of \(3.30 \mathrm{mi} / \mathrm{h}\) at an angle \(62.5^{\circ}\) north of west relative to the water. The river is \(0.505 \mathrm{mi}\) wide and carries an eastward current of \(1.25 \mathrm{mi} / \mathrm{h}\). How far upstream is the boat when it reaches the opposite shore?

Short Answer

Expert verified
The boat is displaced upstream a distance of \(d_{\text{upstream}}\) miles.

Step by step solution

01

Determine the rowboat's component velocities

The rowboat's velocity can be broken down into two components: one parallel to the current (west-east direction), and one perpendicular to the current (south-north direction). Using the given angle \(62.5^{\circ}\) and the velocity \(3.30 \mathrm{mi} / \mathrm{h}\), we can calculate the velocity components with basic trigonometry as follows: \(v_{\text{west}} = 3.30 \, \mathrm{mi/h} \cdot \cos(62.5^{\circ})\), and \(v_{\text{north}} = 3.30 \, \mathrm{mi/h} \cdot \sin(62.5^{\circ})\).
02

Determine effect of the current

The river current affects the west-east velocity of the boat and thus how far upstream it ends up. The westward velocity of the boat relative to the Earth is the difference of the boat's westward velocity and the river's eastward velocity. We find this by subtracting the current's speed from the westward component: \(v_{\text{west, Earth}} = v_{\text{west}} - 1.25 \, \mathrm{mi/h}\). The northward velocity (perpendicular to the current) is unaffected by the current and remains \(v_{\text{north}}\).
03

Calculate the time to cross the river

We can find the time it takes for the boat to cross the river by dividing the width of the river by the northward component of the boat's velocity: \(t = 0.505 \, \mathrm{mi} / v_{\text{north}}\). This gives us the time in hours that the boat takes to cross the river.
04

Find the boat's displacement upstream

We can find how far upstream the boat ends up by multiplying the time to cross the river by the westward velocity of the boat relative to the Earth: \(d_{\text{upstream}} = v_{\text{west, Earth}} \cdot t\). This will give us the boat's displacement upstream in miles.

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Key Concepts

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

Component Velocities
In physics, when an object moves in a plane, its velocity can be broken down into component velocities. These components are often aligned with coordinate axes such as the north-south and east-west directions. This breakdown helps us to understand the movement in each direction separately. For a rowboat moving at an angle, like in our exercise, we use trigonometry to resolve the velocity into two components:
  • Westward Component (\(v_{\text{west}}\)): Calculated using the cosine of the given angle. It is the velocity in the horizontal plane or direction of the river current.
  • Northward Component (\(v_{\text{north}}\)): Calculated using the sine of the given angle. It represents the velocity perpendicular to the river current.
Breaking down the velocities is crucial because it allows us to manage the complexity of motion in two dimensions. We have two simpler one-dimensional motions summed up to understand the real velocity path.
Trigonometry in Physics
Trigonometry plays a key role in physics, particularly when resolving forces and velocities into components. In our scenario, we used trigonometry to split the rowboat's velocity into two perpendicular components. This is achieved by:
  • Cosine Function: Helps to find the adjacent side of the triangle, which in this case, is the westward velocity component. The calculation is given by \(v_{\text{west}} = v \cdot \cos(\theta)\).
  • Sine Function: Used to determine the opposite side, which is the northward velocity. This is calculated by \(v_{\text{north}} = v \cdot \sin(\theta)\).
Employing trigonometric functions makes it fairly straightforward to calculate component values from the total velocity and the angle of motion. Understanding these basics of trigonometry significantly simplifies the analysis of physical systems involving angles.
River Current Effects
A river's current can significantly affect how a boat travels across it. This situation introduces concepts of relative velocity. When the current moves in a specific direction, it adds or reduces the boat's velocity in that same direction. In our exercise:
  • Effect on Westward Velocity: While the rowboat tries to move westward, the eastward current slows it down. The effective velocity relative to the earth is the vector difference, \(v_{\text{west, Earth}} = v_{\text{west}} - 1.25 \; \text{mi/h}\).
  • Northward Velocity: Unaffected by the eastward river current, allowing the rowboat to proceed north at the calculated rate.
The river pushes the boat downstream more than originally intended, affecting the actual point where it reaches the opposite bank. Therefore, factoring in the current's velocity is critical to accurately determine the boat's course and endpoint.

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Most popular questions from this chapter

A small map shows Atlanta to be 730 miles in a direction \(5^{\circ}\) north of east from Dallas. The same map shows that Chicago is 560 miles in a dircction \(21^{\text {n }}\) west of north from Atlanta. Assume a flat Earth and use the given information to find the displacement from Dallas to Chicago.

A home run is hit in such a way that the baseball just clears a wall \(21 \mathrm{~m}\) high, located \(130 \mathrm{~m}\) from home plate. The ball is hit at an angle of \(35^{\circ}\) to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall. and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of \(1.0 \mathrm{~m}\) above the ground.)

A fireman \(50.0 \mathrm{~m}\) away from a burming building directs a stream of water from a ground-level fire hose at an angle of \(30.0^{\circ}\) above the horizontal. If the speed of the stream as it leaves the hose is \(40.0 \mathrm{~m} / \mathrm{s}\), at what height will the stream of water strike the building?

A golfer takes two putts to get his ball into the hole once he is on the green. The first putt displaces the ball \(6.00 \mathrm{~m}\) east, the second \(5.40 \mathrm{~m}\) south. What displacement would have been needed to get the ball into the hole on the first putt?

A hunter wishes to cross a river that is \(1.5 \mathrm{~km}\) wide and flows with a speed of \(5.0 \mathrm{~km} / \mathrm{h}\) parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of \(12 \mathrm{~km} / \mathrm{h}\) with respect to the water. What is the minimum time necessary for crossing?
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