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Chapter 2: Problem 2
Does a speedometer measure speed or velocity?
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The maximum braking acceleration of a car on a dry road is about \(8 \mathrm{m} / \mathrm{s}^{2} .\) If two cars move head-on toward each other at 88 \(\mathrm{km} / \mathrm{h}(55 \mathrm{mi} / \mathrm{h}),\) and their drivers brake when they're \(85 \mathrm{m}\) apart. will they collide? If so, at what relative speed? If not, how far apart will they be when they stop? Plot distance versus time for both cars on a single graph.
Consider an object traversing a distance \(L\), part of the way at speed \(v_{1}\) and the rest of the way at speed \(v_{2} .\) Find expressions for the object's average speed over the entire distance \(L\) when the object moves at each of the two speeds \(v_{1}\) and \(v_{2}\) for (a) half the total time and (b) half the total distance. (c) In which case is the average speed greater?
Starting from home, you bicycle 24 km north in 2.5 h and then turn around and pedal straight home in \(1.5 \mathrm{h}\). What are your (a) displacement at the end of the first \(2.5 \mathrm{h},\) (b) average velocity over the first \(2.5 \mathrm{h},\) (c) average velocity for the homeward leg of the trip, (d) displacement for the entire trip, and (e) average velocity for the entire trip?
You're a consultant on a movie set, and the producer wants a car to drop so that it crosses the camera's field of view in time At. The field of view has height \(h .\) Derive an expression for the height above the top of the field of view from which the car should be released.
An object's position is given by \(x=b t+c t^{3},\) where \(b=1.50 \mathrm{m} / \mathrm{s}, c=0.640 \mathrm{m} / \mathrm{s}^{3},\) and \(t\) is time in seconds. To study the limiting process leading to the instantaneous velocity, calculate the object's average velocity over time intervals from (a) \(1.00 \mathrm{s}\) to \(3.00 \mathrm{s},\) (b) \(1.50 \mathrm{s}\) to \(2.50 \mathrm{s},\) and \((\mathrm{c}) 1.95 \mathrm{s}\) to \(2.05 \mathrm{s}.\) (d) Find the instantaneous velocity as a function of time by differentiating, and compare its value at 2 s with your average velocities.
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