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Chapter 2: Problem 72

You are given the kinematic equation or equations that are used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equation(s). Be sure that the answer your problem requests is consistent with the equation(s) given. b. Draw the pictorial representation for your problem. c. Finish the solution of the problem. $$64 \mathrm{m}=0 \mathrm{m}+(32 \mathrm{m} / \mathrm{s})(4 \mathrm{s}-0 \mathrm{s})+\frac{1}{2} a_{x}(4 \mathrm{s}-0 \mathrm{s})^{2}$$

Short Answer

Expert verified
a) A car starts from rest and accelerates uniformly. After 4 seconds it has covered a distance of 64 meters. b) Pictorial representation described above c) The car's acceleration is \( -8 \, m/s^{2} \). This means the car is decelerating.

Step by step solution

01

Articulating scenario

Consider a scenario where a car starts accelerating from rest at a uniform rate. After 4 seconds, it has travelled a distance of 64 meters. This scenario matches with our given kinematic equation.
02

Pictorial representation

To visually represent the problem, draw a straight horizontal line to represent a flat road. At one end of the line, draw a small square to represent the car at the start (at rest) and mark this position as 0 m. At the other end of the line, mark the position as 64 m. Draw an arrow from the car to this point showing the direction of motion. Along the arrow, indicate that the total time taken for this journey is 4 seconds.
03

Solving for acceleration

The missing variable in our equation is \( a_{x} \), the acceleration of the car. The initial equation \( 64 \mathrm{m}=0 \mathrm{m}+(32 \mathrm{m} / \mathrm{s})(4 \mathrm{s}))+\frac{1}{2} a_{x}(4 \mathrm{s})^{2} \) simplifies to \( 64 = 128 + 8 a_{x} \). You can solve it for \( a_{x} \) by subtracting 128 from both sides, which gives you \( -64 = 8a_{x} \), and then dividing by 8, which provides \( a_{x} = -8 \, m/s^{2} \)

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Key Concepts

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

Acceleration
Acceleration is the rate at which an object's velocity changes over time. In our original problem, the car starts from rest, which means its initial velocity is 0 m/s. As it accelerates, it increases this velocity at a uniform rate. This is represented as \( a_x \) in the kinematic equation.Key considerations in understanding acceleration include:
  • Uniform Acceleration: This is when an object's velocity changes by the same amount every second.
  • Calculating Acceleration: From the equation \( 64 = 128 + 8 a_x \), we solved for acceleration by isolating \( a_x \), resulting in \( a_x = -8 \, m/s^2 \). The negative sign indicates that the car is decelerating because the calculated distance was accounted for from the initial measurement error.
This concept is crucial because acceleration helps us understand how quickly an object reaches its moving speed or how quickly it can stop.
Distance-Time Relationship
The distance-time relationship in physics describes how distance varies over time for moving objects. In our problem, 64 meters is the distance the car travels in 4 seconds, which is precisely what is modeled by our kinematic equation.The kinematic equation applied here is \( d = v_i t + \frac{1}{2} a t^2 \), where:
  • \( v_i \) is the initial velocity (0 m/s for the car at rest).
  • t is the time duration of the motion (4 seconds in the problem).
  • a indicates acceleration.
This shows that distance traveled is a combination of the distance covered by initial velocity and the additional distance covered due to acceleration over time.
Physics Problem Solving
Solving physics problems often involves applying mathematical formulas to real-world scenarios, imagining the situation, and solving for unknowns. The exercise presented a real-world scenario: a car accelerating from rest.To solve problems like this effectively:
  • Identify given values and assign them to variables (e.g., initial velocity \( v_i \), time \( t \), and distance \( d \)).
  • Select the correct equation that incorporates these known values and the unknown variable you need to solve for, in this case, \( a \), acceleration.
  • Solve the equation step by step, isolating the unknown variable. As shown, \( 64 = 128 + 8 a_x \) translates to \( a_x = -8 \, m/s^2 \) when solved.
A systematic approach aids in solving physics problems logically and effectively.
Pictorial Representation
Creating a pictorial representation is a valuable step when solving physics problems. It translates abstract information into a visual format, making the problem easier to understand. In our exercise, envision a straight horizontal line representing the road:
  • Mark the initial position (0 m) and final position (64 m).
  • Draw an arrow from the car's start to its end, indicating direction and motion.
  • Label all known values, such as distance and time (4 seconds), along the arrow.
This visual aid helps in conceptualizing the problem, ensuring all elements align with the kinematic equations being applied. Visuals support comprehension, allowing one to see the entire scheme and check that calculations and assumptions align.

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Most popular questions from this chapter

The minimum stopping distance for a car traveling at a speed of \(30 \mathrm{m} / \mathrm{s}\) is \(60 \mathrm{m},\) including the distance traveled during the driver's reaction time of \(0.50 \mathrm{s}\) a. What is the minimum stopping distance for the same car traveling at a speed of \(40 \mathrm{m} / \mathrm{s} ?\) b. Draw a position-versus-time graph for the motion of the car in part a. Assume the car is at \(x_{0}=0\) m when the driver first sees the emergency situation ahead that calls for a rapid halt.

Alan leaves Los Angeles at 8: 00 a.m. to drive to San Francisco, 400 mi away. He travels at a steady 50 mph. Beth leaves Los Angeles at 9: 00 a.m. and drives a steady 60 mph. a. Who gets to San Francisco first? b. How long does the first to arrive have to wait for the second?

Careful measurements have been made of Olympic sprinters in the 100 -meter dash. A quite realistic model is that the sprinter's velocity is given by $$v_{x}=a\left(1-e^{-b t}\right)$$ where \(t\) is in \(\mathrm{s}, v_{x}\) is in \(\mathrm{m} / \mathrm{s},\) and the constants \(a\) and \(b\) are characteristic of the sprinter. Sprinter Carl Lewis's run at the 1987 World Championships is modeled with \(a=11.81 \mathrm{m} / \mathrm{s}\) and \(b=0.6887 s^{-1}\) a. What was Lewis's acceleration at \(t=0 \mathrm{s}, 2.00 \mathrm{s},\) and \(4.00 \mathrm{s} ?\) b. Find an expression for the distance traveled at time \(t\) c. Your expression from part \(\mathbf{b}\) is a transcendental equation, meaning that you can't solve it for \(t .\) However, it's not hard to use rial and error to find the time needed to travel a specific distance. To the nearest \(0.01 \mathrm{s}\), find the time Lewis needed to sprint \(100.0 \mathrm{m} .\) His official time was \(0.01 \mathrm{s}\) more than your answer, showing that this model is very good, but not perfect.

A jet plane is cruising at 300 m/s when suddenly the pilot turns the engines up to full throttle. After traveling \(4.0 \mathrm{km}\), the jet is moving with a speed of \(400 \mathrm{m} / \mathrm{s}\) a. What is the jet's acceleration, assuming it to be a constant acceleration? b. Is your answer reasonable? Explain.

Bob is driving the getaway car after the big bank robbery. He's going \(50 \mathrm{m} / \mathrm{s}\) when his headlights suddenly reveal a nail strip that the cops have placed across the road \(150 \mathrm{m}\) in front of him. If Bob can stop in time, he can throw the car into reverse and escape. But if he crosses the nail strip. all his tires will go flat and he will be caught. Bob's reaction time before he can hit the brakes is \(0.60 \mathrm{s}\), and his car's maximum deceleration is \(10 \mathrm{m} / \mathrm{s}^{2} .\) Is Bob in jail?
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