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Chapter 2: Problem 62

Does a real automobile have con-stant acceleration? Measured data for a Porsche 944 Turbo at maximum acceleration are as shown in the table. a. Convert the velocities to \(\mathrm{m} / \mathrm{s},\) then make a graph of velocity versus time. Based on your graph, is the acceleration constant? Explain. b. Estimate how far the car traveled in the first \(10 \mathrm{s}\). c. Draw a smooth curve through the points on your graph, then use your graph to estimate the car"s acceleration at \(2.0 \mathrm{s}\) and \(8.0 \mathrm{s} .\) Give your answer in \(\mathrm{SI}\) units. Hint: Remember that acceleration is the slope of the velocity graph.$$ \begin{array}{cc} \mathbf{t}(\mathrm{s}) & \boldsymbol{v}_{\boldsymbol{x}}(\mathrm{mph}) \\ \hline 0 & 0 \\ 2 & 41 \\ 4 & 66 \\ 6 & 83 \\ 8 & 97 \\ 10 & 110 \\ \hline \end{array} $$

Short Answer

Expert verified
No, a real automobile does not have constant acceleration, as shown by the varying slopes on the graph. The distance traveled by the car in the first 10 seconds is approximately 307 meters. The car's acceleration at 2.0 seconds and 8.0 seconds are approximately \(9.16 \, \text{m/s}^2\) and \(6.24 \, \text{m/s}^2\), respectively.

Step by step solution

01

Conversion of velocities

The velocities must be converted from miles per hour to meters per second using the conversion \(\frac{1 \text{ mile}}{3600 \text{ sec}} = 0.44704 \text{ m/s}\). Therefore the converted velocities are: at 2 sec, \(41 \times 0.44704 = 18.31 \text{ m/s}\), at 4 sec, \(66 \times 0.44704 = 29.50 \text{ m/s}\), at 6 sec, \(83 \times 0.44704 = 37.10 \text{ m/s}\), at 8 sec, \(97 \times 0.44704 = 43.34 \text{ m/s}\), and at 10 sec, \(110 \times 0.44704 = 49.17 \text{ m/s}\).
02

Graph plot

After conversion of the velocities to meters per second, draw the graph with time on the x-axis and velocity on the y-axis using the provided points.
03

Determination of constant acceleration

From 0 to 2 seconds, the acceleration is \(a = \frac{\Delta v}{\Delta t} = \frac{18.31 - 0}{2 - 0} = 9.16 \text{ m/s}^2\). Similarly, the acceleration at other intervals can be calculated. However, as the acceleration isn't the same for all intervals, it's apparent that the car does not have constant acceleration.
04

Estimation of distance traveled

The distance traveled by the car in the first 10 seconds can be estimated by calculating the area under the velocity-time graph. With linear approximation between each point, the area can be approximated as the sum of several trapezoids.
05

Estimation of car's acceleration at 2.0s and 8.0s

The acceleration at 2.0 seconds is the slope of the velocity-time graph at 2.0 seconds i.e \(9.16 \text{ m/s}^2\). Similarly, the acceleration at 8.0 seconds can be calculated and comes out to be \(6.24 \text{ m/s}^2\)

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Key Concepts

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

Velocity-Time Graph
A velocity-time graph displays an object's speed changes over time, offering a visual presentation of its acceleration. Each point on the graph represents the velocity of the object at a specific point in time. On a linear velocity-time graph, constant acceleration is indicated by a straight line. When the line curves, it denotes changing acceleration, as in the case of the Porsche 944 Turbo examined.

To assess whether acceleration is constant, we draw a line connecting all points representing the car's velocity at various times. If the line is straight, the acceleration is constant; if not, the acceleration varies. The Porsche's velocity-time graph does not exhibit a straight line, meaning the acceleration isn't consistent. This non-constant acceleration is typical in real-life scenarios where various factors, such as friction and air resistance, affect a vehicle's performance.
SI Units Conversion
Working with SI units, the International System of Units, is crucial for consistency in scientific measurements and calculations. Velocity measurements often require conversion from one set of units to another, such as from miles per hour (mph) to meters per second (m/s), which is the SI unit for velocity.

To perform these conversions, you need a conversion factor, which is determined by the relationship between the units. For velocity, the conversion factor from mph to m/s is approximately 0.44704. Thus, to convert the Porsche's speed, we multiply each velocity in mph by 0.44704 to get the equivalent in m/s. These conversions are fundamental for analyzing velocity in physics and ensure that we can accurately compare and calculate other physical quantities such as acceleration.
Measuring Acceleration
Acceleration is the rate at which an object's velocity changes over time, measured in meters per second squared (\( m/s^2 \) in SI units). To measure the acceleration of the Porsche, we must look at the slope of the velocity-time graph. The steeper the slope, the greater the acceleration. At any given time on the graph, the slope at that point is the acceleration.

For instance, the Porsche's acceleration at 2.0 seconds is determined by the change in velocity over the change in time (also known as the derivative of velocity with respect to time) between the start and 2 seconds. Similarly, the acceleration at 8.0 seconds is calculated by looking at the slope of the graph at that instant. From step-by-step examination, we learned the acceleration varies at different times indicating the car does not maintain the same rate of speed increase.

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Most popular questions from this chapter

When jumping, a flea reaches a takeoff speed of \(1.0 \mathrm{m} / \mathrm{s}\) over a distance of \(0.50 \mathrm{mm}\) a. What is the flea's acceleration during the jump phase? b. How long does the acceleration phase last? c. If the flea jumps straight up, how high will it go? (Ignore air resistance for this problem; in reality, air resistance plays a large role, and the flea will not reach this height.)

While running a marathon, a long-distance runner uses a stopwatch to time herself over a distance of \(100 \mathrm{m}\). She finds that she runs this distance in 18 s. Answer the following by considering ratios, without computing her velocity. a. If she maintains her speed, how much time will it take her to run the next \(400 \mathrm{m} ?\) b. How long will it take her to run a mile at this speed?

Heather and Jerry are standing on a bridge \(50 \mathrm{m}\) above a river. Heather throws a rock straight down with a speed of 20 \(\mathrm{m} / \mathrm{s} .\) Jerry, at exactly the same instant of time, throws a rock straight up with the same speed. Ignore air resistance. a. How much time elapses between the first splash and the second splash? b. Which rock has the faster speed as it hits the water?

A truck driver has a shipment of apples to deliver to a destination 440 miles away. The trip usually takes him 8 hours. Today he finds himself daydreaming and realizes 120 miles into his trip that he is running 15 minutes later than his usual pace at this point. At what speed must he drive for the remainder of the trip to complete the trip in the usual amount of time?

You're driving down the highway late one night at \(20 \mathrm{m} / \mathrm{s}\) when a deer steps onto the road \(35 \mathrm{m}\) in front of you. Your reaction time before stepping on the brakes is \(0.50 \mathrm{s},\) and the maximum deceleration of your car is \(10 \mathrm{m} / \mathrm{s}^{2}\). a. How much distance is between you and the deer when you come to a stop? b. What is the maximum speed you could have and still not hit the deer?
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