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Chapter 2: Problem 1
Under what conditions are average and instantaneous velocity equal?
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An object starts moving in a straight line from position \(x_{0}\), at time \(t=0,\) with velocity \(v_{0} .\) Its acceleration is given by \(a=a_{0}+b t\) where \(a_{0}\) and \(b\) are constants. Use integration to find expressions for (a) the instantaneous velocity and (b) the position, as functions of time.
A racing car undergoing constant acceleration covers \(140 \mathrm{~m}\) in \(3.6 \mathrm{~s}\). (a) If it's moving at \(53 \mathrm{~m} / \mathrm{s}\) at the end of this interval, what was its speed at the beginning of the interval? (b) How far did it travel from rest to the end of the \(140-\mathrm{m}\) distance?
A model rocket is launched straight upward. Its altitude \(y\) as a function of time is given by \(y=b t-c t^{2},\) where \(b=82 \mathrm{m} / \mathrm{s}, c=4.9 \mathrm{m} / \mathrm{s}^{2}, t\) is the time in seconds, and \(y\) is in meters. (a) Use differentiation to find a general expression for the rocket's velocity as a function of time. (b) When is the velocity zero?
A castle's defenders throw rocks down on their attackers from a 15-m-high wall, with initial speed 10 m/s. How much faster are the rocks moving when they hit the ground than if they were simply dropped?
You check your odometer at the beginning of a day's driving and again at the end. Under what conditions would the difference between the two readings represent your displacement?
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