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Chapter 2: Problem 18
What's the conversion factor from meters per second to miles per hour?
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In a drag race, the position of a car as a function of time is given by \(x=b t^{2},\) with \(b=2.000 \mathrm{m} / \mathrm{s}^{2} .\) In an attempt to determine the car's velocity midway down a 400 -m track, two observers stand at the 180 -m and 220 -m marks and note when the car passes. (a) What value do the two observers compute for the car's velocity over this 40 -m stretch? Give your answer to four significant figures. (b) By what percentage does this observed value differ from the instantaneous value at \(x=200 \mathrm{m} ?\)
Consider two possible definitions of average speed: (a) the average of the values of the instantaneous speed over a time interval and (b) the magnitude of the average velocity. Are these definitions equivalent? Give two examples to demonstrate your conclusion.
The standard 26-mile, 385-yard marathon dates to 1908, when the Olympic marathon started at Windsor Castle and finished before the Royal Box at London's Olympic Stadium. Today's top marathoners achieve times around 2 hours, 3 minutes for the standard marathon. (a) What's the average speed of a marathon run in this time? (b) Marathons before 1908 were typically about 25 miles. How much longer does the race last today as a result of the extra mile and 385 yards, assuming it's run at part (a)'s average speed?
During the complicated sequence that landed the rover Curiosity on Mars in 2012, the spacecraft reached an altitude of \(142 \mathrm{m}\) above the Martian surface, moving vertically downward at \(32.0 \mathrm{m} / \mathrm{s}\). It then entered a so-called constant deceleration (CD) phase, during which its velocity decreased steadily to \(0.75 \mathrm{m} / \mathrm{s}\) while it dropped to an altitude of \(23 \mathrm{m}\). What was the magnitude of the spacecraft's acceleration during this CD phase?
An object's acceleration is given by the expression \(a(t)=-a_{0} \cos \omega t,\) where \(a_{0}\) and \(\omega\) are positive constants. Find expressions for the object's (a) velocity and (b) position as functions of time. Assume that at time \(t=0\) it starts from rest at its greatest positive displacement from the origin. (c) Determine the magnitudes of the object's maximum velocity and maximum displacement from the origin.
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