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Chapter 4: Problem 7

A car accelerates from rest to a final constant velocity \(V_{f}\) and a police officer records the following velocities at various locations \(x\) along the highway: $$\begin{array}{ll} x=0 & V=0 \mathrm{mph} \\ x=100 \mathrm{ft} & V=34.8 \mathrm{mph} \\ x=200 \mathrm{ft} & V=47.6 \mathrm{mph} \\ x=300 \mathrm{ft} & V=52.3 \mathrm{mph} \\ x=400 \mathrm{ft} & V=54.0 \mathrm{mph} \\ x=1000 \mathrm{ft} & V=55.0 \mathrm{mph}=V_{f} \end{array}$$ Find a mathematical Eulerian expression for the velocity \(V\) traveled by the car as a function of the final velocity \(V_{f}\) of the car and \(x\) if \(t=0\) at \(V=0 .[\) Hint: Try an exponential fit to the data.

Short Answer

Expert verified
The mathematical expression for the velocity travelled by the car as a function of the final velocity \(V_f\) of the car and distance \(x\) is \(V = V_f(1 - e^{-0.02x})\).

Step by step solution

01

Formulate the Exponential Function

Based on the characteristics of the velocity and the hint given, we will model the velocity with an exponential decay function of the form \(V = V_f - Ae^{-Bx}\). Here, \(V_f\) is the final constant velocity, \(A\) and \(B\) are constants to be determined, \(e\) is the mathematical constant (approx. 2.718), and \(x\) is the distance.
02

Computing Constant A

At \(x = 0\), the velocity \(V = 0\). Plugging these values into our equation gives us \(0 = V_f - A\). Therefore, \(A = V_f\). The velocity function then becomes \(V = V_f(1 - e^{-Bx})\).
03

Computing Constant B

We can find \(B\) by considering the velocity when \(x = 100\) ft. Plugging these values into the equation gives us \(34.8 = 55(1 - e^{-100B})\). Solving this for \(B\) gives us an approximate value of \(B = 0.02\) per foot.
04

Final Function

Substituting \(A = V_f\) and \(B = 0.02\) per foot into our function from Step 1, we obtain the final mathematical expression for the velocity: \(V = V_f(1 - e^{-0.02x})\). This function allows the calculation of the velocity at any point \(x\) along the highway given the final velocity \(V_f\).

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Most popular questions from this chapter

A bicyclist leaves from her home at 9 A.M. and rides to a beach \(40 \mathrm{mi}\) away. Because of a breeze off the ocean, the temperature at the beach remains \(60^{\circ} \mathrm{F}\) throughout the day. At the cyclist's home the temperature increases linearly with time, going from \(60^{\circ} \mathrm{F}\) at 9 A.M. to \(80^{\circ} \mathrm{F}\) by 1 P.M. The temperature is assumed to vary linearly as a function of position between the cyclist's home and the beach. Determine the rate of change of temperature observed by the cyclist for the following conditions: (a) as she pedals 10 mph through a town 10 mi from her home at 10 A.M.; (b) as she eats lunch at a rest stop \(30 \mathrm{mi}\) from her home at noon; (c) as she arrives enthusiastically at the beach at 1 P.M., pedaling 20 mph.

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