/*! 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 30 A speedboat increases its speed ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 30

A speedboat increases its speed uniformly from \(v_{i}=\) \(20.0 \mathrm{~m} / \mathrm{s}\) to \(v_{f}=30.0 \mathrm{~m} / \mathrm{s}\) in a distance of \(2.00 \times 10^{2} \mathrm{~m}\). (a) Draw a coordinate system for this situation and label the relevant quantities, including vectors. (b) For the given information, what single equation is most appropriate for finding the acceleration? (c) Solve the equation selected in part (b) symbolically for the boat's acceleration in terms of \(v_{f}, v_{i}\), and \(\Delta x\). (d) Substitute given values, obtaining that acceleration. (e) Find the time it takes the boat to travel the given distance.

Short Answer

Expert verified
The acceleration of the boat is \(2.5 m/s^2\), and it takes \(4 s\) for the boat to travel the given distance.

Step by step solution

01

Sketch the Situation

Draw a speed-versus-time graph, label the initial velocity \(v_{i}=20.0 \, m/s\), final velocity \(v_{f}=30.0 \, m/s\), and the distance traveled \(\Delta x = 200 \, m\). The relevant quantities are all scalars, so they will each have a magnitude but no direction.
02

Determine Appropriate Equation

To find the acceleration, we can use the following kinematic equation, which allows us to solve for acceleration when we know initial and final velocities, and displacement: \(v_f^2 = v_i^2 + 2a\Delta x\)
03

Solve the Equation Symbolically

Rearranging the equation, we can isolate \(a\) to solve for it symbolically: \(a = (v_f^2 - v_i^2) / (2\Delta x)\)
04

Substitute Given Values

Substitute \(v_i = 20 m/s\), \(v_f = 30 m/s\), and \(\Delta x = 200 m\) into the equation: \(a = (30^2 - 20^2) / (2 * 200) m/s^2 = 2.5 m/s^2\)
05

Find the Time it Takes

We can use the equation \(v_f = v_i + at\) to find the time it takes for the boat to travel the given distance. Rearranging for \(t\): \(t = (v_f - v_i) / a\). Substitute \(v_i =20 m/s\), \(v_f = 30 m/s\), and \(a = 2.5 m/s^2\) into the equation to find \(t = (30 - 20) / 2.5 s = 4 s\)

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Key Concepts

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

Uniform Acceleration
Uniform acceleration occurs when an object's velocity changes at a constant rate. This is a fundamental concept in kinematics, the branch of physics that deals with motion. Imagine you're in a car that starts to speed up; if the car's speed increases by the same amount every second, then the car has uniform acceleration. Mathematically, it's expressed with the equation \( a = \frac{\Delta v}{\Delta t} \) where \( \Delta v \) is the change in velocity, and \( \Delta t \) is the time period over which the change occurs. This simplicity allows us to predict the future position and velocity of objects under such acceleration with great accuracy.
Final Velocity
Final velocity (\( v_f \)) is the speed of an object at the end of a given time interval. When dealing with motion under uniform acceleration, the final velocity can be determined from the initial velocity, the acceleration, and the time taken. It's formulated as \( v_f = v_i + at \), where \( v_i \) is the initial velocity and \( a \) is the acceleration. In the context of our speedboat example, the final velocity tells us how fast the boat is moving after it has accelerated for a certain distance.
Initial Velocity
Initial velocity (\( v_i \)) is where the story of motion begins; it's the velocity of the object before any acceleration has been applied. It serves as the starting point for calculations within kinematic equations. When you solve problems involving moving objects, knowing the initial velocity is vital since it sets the stage for how the object will behave as forces are applied to it. Understanding both initial and final velocities is key to analyzing motion.
Displacement
Displacement (\( \Delta x \)) represents the change in position of an object. It is a vector quantity, which means it has both a magnitude and a direction. In the context of uniform acceleration, displacement tells us how far an object has traveled while accelerating from its initial to its final velocity. The kinematic equation that includes displacement is \( v_f^2 = v_i^2 + 2a\Delta x \), revealing the relationship between displacement, velocities, and acceleration. Displacement is especially important in calculating other aspects of motion, such as acceleration in our speedboat example.
Speed-Versus-Time Graph
A speed-versus-time graph is an excellent tool for visualizing an object's motion over time. The slope of a speed-versus-time graph represents acceleration. When acceleration is uniform, the graph is a straight line, and the slope is constant. In the speedboat problem, such a graph would show a straight, sloping line representing the boat's increasing speed over time. The area under this line would give us the displacement of the boat, a connection that is not only mathematical but also visual, helping students better understand the relationship between speed, time, and distance traveled.

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Most popular questions from this chapter

A jet plane lands with a speed of \(100 \mathrm{~m} / \mathrm{s}\) and can accelerate at a maximum rate of \(-5.00 \mathrm{~m} / \mathrm{s}^{2}\) as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest? (b) Can this plane land on a small tropi\(\mathrm{cal}\) island airport where the runway is \(0.800 \mathrm{~km}\) long?

Two boats start together and race across a \(60-\mathrm{km}\)-wide lake and back. Boat A goes across at \(60 \mathrm{~km} / \mathrm{h}\) and returns at \(60 \mathrm{~km} / \mathrm{h}\). Boat B goes across at \(30 \mathrm{~km} / \mathrm{h}\), and its crew, realizing how far behind it is getting, returns at \(90 \mathrm{~km} / \mathrm{h}\). Turnaround times are negligible, and the boat that completes the round trip first wins. (a) Which boat wins and by how much? (Or is it a tie?) (b) What is the average velocity of the winning boat?

A certain freely falling object, released from rest, requires \(1.50 \mathrm{~s}\) to travel the last \(30.0 \mathrm{~m}\) before it hits the ground. (a) Find the velocity of the object when it is \(30.0 \mathrm{~m}\) above the ground. (b) Find the total distance the object travels during the fall.

A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of \(12 \mathrm{~m} / \mathrm{s}\), skates by with the puck. After \(3.0 \mathrm{~s}\), the first player makes up his mind to chase his opponent. If he accelerates uniformly at \(4.0 \mathrm{~m} / \mathrm{s}^{2}\), (a) how long does it take him to catch his opponent, and (b) how far has he traveled in that time? (Assume the player with the puck remains in motion at constant speed.)

A ball is thrown vertically upward with a speed of 25.0 \(\mathrm{m} / \mathrm{s}\). (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started?
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