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Chapter 2: Problem 30

The position of a particle moving along the \(x\) -axis is given by \(x=\left(11+14 t-2.0 t^{2}\right),\) where \(t\) is in seconds and \(x\) is in meters. What is the average velocity during the time interval from \(t=1.0 \mathrm{~s}\) to \(t=4.0 \mathrm{~s} ?\)

Short Answer

Expert verified
Answer: The average velocity during the time interval from \(t=1.0\text{ s}\) to \(t=4.0\text{ s}\) is \(4.0 \frac{\text{m}}{\text{s}}\).

Step by step solution

01

Calculate Position at the Starting and Ending of the Time Interval

Using the position function, find the position of the particle at \(t=1.0s\) and \(t=4.0s\): \(x(1.0) = 11 + 14(1.0) - 2.0(1.0)^2\) \(x(4.0) = 11 + 14(4.0) - 2.0(4.0)^2\)
02

Compute Positions

Calculate the positions at the given times: \(x(1.0) = 11 + 14 - 2 = 23\text{ m}\) \(x(4.0) = 11 + 56 - 32 = 35\text{ m}\)
03

Find Displacement

Subtract the initial position from the final position to find the displacement: \(\Delta x = x(4.0) - x(1.0) = 35 - 23 = 12\text{ m}\)
04

Calculate Time Interval

Find the time interval between the given times: \(\Delta t = 4 - 1 = 3\text{ s}\)
05

Compute Average Velocity

Divide the displacement by the time interval to find the average velocity: \(v_\text{avg} = \frac{\Delta x}{\Delta t} = \frac{12}{3} = 4.0 \frac{\text{m}}{\text{s}}\) The average velocity during the time interval from \(t=1.0\text{ s}\) to \(t=4.0\text{ s}\) is \(4.0 \frac{\text{m}}{\text{s}}\).

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Most popular questions from this chapter

In 2005, Hurricane Rita hit several states in the southern United States. In the panic to escape her wrath, thousands of people tried to flee Houston, Texas by car. One car full of college students traveling to Tyler, Texas, 199 miles north of Houston, moved at an average speed of \(3.0 \mathrm{~m} / \mathrm{s}\) for one-fourth of the time, then at \(4.5 \mathrm{~m} / \mathrm{s}\) for another one-fourth of the time, and at \(6.0 \mathrm{~m} / \mathrm{s}\) for the remainder of the trip. a) How long did it take the students to reach their destination? b) Sketch a graph of position versus time for the trip.

The vertical position of a ball suspended by a rubber band is given by the equation $$ y(t)=(3.8 \mathrm{~m}) \sin (0.46 t / \mathrm{s}-0.31)-(0.2 \mathrm{~m} / \mathrm{s}) t+5.0 \mathrm{~m} $$ a) What are the equations for velocity and acceleration for this ball? b) For what times between 0 and \(30 \mathrm{~s}\) is the acceleration zero?

A double speed trap is set up on a freeway. One police cruiser is hidden behind a billboard, and another is some distance away under a bridge. As a sedan passes by the first cruiser, its speed is measured to be \(105.9 \mathrm{mph}\). Since the driver has a radar detector, he is alerted to the fact that his speed has been measured, and he tries to slow his car down gradually without stepping on the brakes and alerting the police that he knew he was going too fast. Just taking the foot off the gas leads to a constant deceleration. Exactly 7.05 s later the sedan passes the second police cruiser. Now its speed is measured to be only \(67.1 \mathrm{mph}\), just below the local freeway speed limit. a) What is the value of the deceleration? b) How far apart are the two cruisers?

A ball is thrown straight upward in the air at a speed of \(15.0 \mathrm{~m} / \mathrm{s} .\) Ignore air resistance. a) What is the maximum height the ball will reach? b) What is the speed of the ball when it reaches \(5.00 \mathrm{~m} ?\) c) How long will it take to reach \(5.00 \mathrm{~m}\) above its initial position on the way up? d) How long will it take to reach \(5.00 \mathrm{~m}\) above its initial position on its way down?

A ball is dropped from the roof of a building. It hits the ground and it is caught at its original height 5.0 s later. a) What was the speed of the ball just before it hits the ground? b) How tall was the building? c) You are watching from a window \(2.5 \mathrm{~m}\) above the ground. The window opening is \(1.2 \mathrm{~m}\) from the top to the bottom. At what time after the ball was dropped did you first see the ball in the window?
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