/*! 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 31 A cart is driven by a large prop... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 31

A cart is driven by a large propeller or fan, which can accelerate or decelerate the cart. The cart starts out at the position \(x=0 \mathrm{m}\) , with an initial velocity of \(+5.0 \mathrm{m} / \mathrm{s}\) and a constant acceleration due to the fan. The direction to the right is positive. The cart reaches a maximum position of \(x=+12.5 \mathrm{m},\) where it begins to travel in the negative direction. Find the acceleration of the cart.

Short Answer

Expert verified
The cart's acceleration is \( -1.0 \mathrm{m/s^2} \).

Step by step solution

01

Identify Known Values

We know the initial position is \( x_0 = 0 \mathrm{m} \), the initial velocity is \( v_0 = +5.0 \mathrm{m/s} \), and the maximum position is \( x = +12.5 \mathrm{m} \). At this point, the cart changes direction implying the velocity is zero and the acceleration is constant.
02

Apply Kinematic Equation for Maximum Position

Use the kinematic equation: \[ v^2 = v_0^2 + 2a(x - x_0) \]Where \( v = 0 \mathrm{m/s} \) at the maximum position. Substitute the known values:\[ 0 = (5.0)^2 + 2a(12.5 - 0) \]
03

Solve for Acceleration

Rearrange the equation to solve for \( a \):\[ 0 = 25 + 25a \]Simplify and solve for \( a \):\[ a = -\frac{25}{25} = -1.0 \mathrm{m/s^2} \]
04

Interpret the Result

The negative sign of acceleration means the cart is decelerating as it moves to the maximum position at \( +12.5 \mathrm{m} \).

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Key Concepts

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

Initial Velocity
Initial velocity is a vital concept in kinematics, representing the speed and direction of a moving object at the start of observing its motion. In our exercise, the cart begins with an initial velocity of \(+5.0 \mathrm{m/s}\), indicating it is moving to the right.
  • The positive sign means the motion is in the positive direction, usually towards the right.
  • Initial velocity is crucial for predicting how an object's position changes over time.
Understanding initial velocity helps in computing future positions and velocities, making it a cornerstone in solving kinematic problems, particularly when calculating motions under the influence of forces like gravity or, in this case, a fan propeller.
Constant Acceleration
Constant acceleration refers to an unchanging acceleration experienced by an object over time. In our scenario, the cart experiences a constant acceleration due to the fan's force. This means that every second, the velocity of the cart changes by a fixed amount.
  • Constant acceleration simplifies calculations because it allows the use of kinematic equations, which assume unvarying acceleration.
In the exercise, the cart's acceleration is found to be \(-1.0 \mathrm{m/s^2}\). The negative value indicates that the acceleration is in the opposite direction of the initial velocity, which causes the cart to slow down as it approaches the maximum position. This deceleration eventually brings the cart to a stop before reversing its direction.
Position
Position in physics refers to the specific point where an object is located at any given time on a linear path.
  • In this problem, position is given as \(x=0 \mathrm{m}\) initially and \(x=+12.5 \mathrm{m}\) when the cart changes direction.
It serves as a reference point to measure how far and in which direction an object has moved. The cart began at \(0 \mathrm{m}\) and traveled to \(12.5 \mathrm{m}\), indicating a positive displacement. This maximum position tells us when the velocity reached zero and the cart began moving in the opposite direction, which is crucial for finding the acceleration.
Kinematic Equations
Kinematic equations are essential tools for solving problems involving motion with constant acceleration. These mathematical formulas relate velocity, acceleration, time, and displacement. In this exercise, the relevant kinematic equation is:\[ v^2 = v_0^2 + 2a(x - x_0) \]This equation is used to find unknown variables like acceleration when other quantities are known.
  • The equation includes the squares of the final and initial velocities, the acceleration, and the change in position.
By substituting the known values into this equation, we are able to solve for the acceleration and conclude that it's \(-1.0 \mathrm{m/s^2}\). The kinematic equations provide a methodical way to find a missing variable, ensuring consistent solutions in physics problems involving motion.

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Most popular questions from this chapter

You are on a train that is traveling at 3.0 \(\mathrm{m} / \mathrm{s}\) along a level straight track. Very near and parallel to the track is a wall that slopes upward at a \(12^{\circ}\) angle with the horizontal. As you face the window \((0.90 \mathrm{m} \text { high, } 2.0 \mathrm{m}\) wide ) in your compartment, the train is moving to the left, as the drawing indicates. The top edge of the wall first appears at window corner A and eventually disappears at window corner \(\mathrm{B}\) . How much time passes between appearance and disappearance of the upper edge of the wall?

In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints to a speed of 6.0 \(\mathrm{m} / \mathrm{s}\) in 1.5 \(\mathrm{s}\) . Assuming that the player accelerates uniformly, determine the distance he runs.

In 1954 the English runner Roger Bannister broke the four-minute barrier for the mile with a time of \(3 : 59.4 \mathrm{s}(3 \mathrm{min} \text { and } 59.4 \mathrm{s}) .\) In 1999 the Moroccan runner Hicham el-Guerrouj set a record of \(3 : 43.13\) s for the mile. If these two runners had run in the same race, each running the entire race at the average speed that earned him a place in the record books, el-Guerrouj would have won. By how many meters?

A tourist being chased by an angry bear is running in a straight line toward his car at a speed of 4.0 \(\mathrm{m} / \mathrm{s}\) . The car is a distance \(d\) away. The bear is 26 \(\mathrm{m}\) behind the tourist and running at 6.0 \(\mathrm{m} / \mathrm{s}\) . The tourist reaches the car safely. What is the maximum possible value for \(d ?\)

A bicyclist makes a trip that consists of three parts, each in the same direction (due north) along a straight road. During the first part, she rides for 22 minutes at an average speed of 7.2 \(\mathrm{m} / \mathrm{s}\) . During the second part, she rides for 36 minutes at an average speed of 5.1 \(\mathrm{m} / \mathrm{s}\) . Finally, during the third part, she rides for 8.0 minutes at an average speed of 13 \(\mathrm{m} / \mathrm{s}\) . (a) How far has the bicyclist traveled during the entire trip? is her average velocity for the trip?
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