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Chapter 2: Problem 38

A car accelerates uniformly from rest to a speed of \(40.0 \mathrm{mi} / \mathrm{h}\) in \(12.0 \mathrm{~s}\). Find (a) the distance the car travels during this time and (b) the constant acceleration of the car.

Short Answer

Expert verified
The car travels approximately 107.28 meters, and the constant acceleration of the car is approximately 1.49 m/s^2.

Step by step solution

01

- Convert speed to m/s

The speed is given in mi/hr but our calculations demand that it be in m/s. The conversion factor is 1 mi/hr = 0.44704 m/s. Hence, \( 40 \mathrm{mi/hr} \) converts to \( 40 \tx{0.44704 m/s} = 17.88 \mathrm{m/s} \).
02

- Find the distance using the equation of motion

For uniform acceleration from rest, the distance traveled is given by the equation \( x = ut + \frac{1}{2} a t^{2} \), where \( x \) is the distance, \( u \) is the initial speed, \( a \) is the acceleration, and \( t \) is the time. Here, the car starts from rest, so the initial speed \( u = 0 \). Therefore, the equation simplifies to \( x = \frac{1}{2} a t^{2} \). We know that \( t = 12 \mathrm{s} \) but the value of \( a \) is not known yet. So, we'll need to find \( a \) before we can calculate \( x \).
03

- Find the acceleration

Acceleration is given by the equation \( a = \frac{(v - u)}{t} \), where \( v \) is final velocity, \( u \) is initial velocity, and \( t \) is time. Here, \( v = 17.88 \mathrm{m/s} \), \( u = 0 \), and \( t = 12.0 \) seconds. Substituting these values in the equation, we have \( a = \frac{(17.88 - 0)}{12} = 1.49 \mathrm{m/s^2} \).
04

- Calculate the distance

Now that we have obtained the acceleration \( a = 1.49 \mathrm{m/s^2} \), we can substitute this value in the simplified equation from step 2 to find the distance: \( x = \frac{1}{2} a t^{2} = 0.5 \tx{1.49 m/s^2} \tx{(12 s)^2} = 107.28 \mathrm{m} \).

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Most popular questions from this chapter

A speedboat increases its speed uniformly from \(v_{i}=\) \(20.0 \mathrm{~m} / \mathrm{s}\) to \(v_{f}=30.0 \mathrm{~m} / \mathrm{s}\) in a distance of \(2.00 \times 10^{2} \mathrm{~m}\). (a) Draw a coordinate system for this situation and label the relevant quantities, including vectors. (b) For the given information, what single equation is most appropriate for finding the acceleration? (c) Solve the equation selected in part (b) symbolically for the boat's acceleration in terms of \(v_{f}, v_{i}\), and \(\Delta x\). (d) Substitute given values, obtaining that acceleration. (e) Find the time it takes the boat to travel the given distance.

A jet plane lands with a speed of \(100 \mathrm{~m} / \mathrm{s}\) and can accelerate at a maximum rate of \(-5.00 \mathrm{~m} / \mathrm{s}^{2}\) as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest? (b) Can this plane land on a small tropi\(\mathrm{cal}\) island airport where the runway is \(0.800 \mathrm{~km}\) long?

One athlete in a race running on a long, straight track with a constant speed \(v_{1}\) is a distance \(d\) behind a second athlete running with a constant speed \(v_{2}\). (a) Under what circumstances is the first athlete able to overtake the second athlete? (b) Find the time \(t\) it takes the first athlete to overtake the second athlete, in terms of \(d, v_{1}\), and \(v_{2}\). (c) At what minimum distance \(d_{2}\) from the leading athlete must the finish line be located so that the trailing athlete can at least tie for first place? Express \(d_{2}\) in terms of \(d, v_{1}\), and \(v_{2}\) by using the result of part (b).

A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of \(12 \mathrm{~m} / \mathrm{s}\), skates by with the puck. After \(3.0 \mathrm{~s}\), the first player makes up his mind to chase his opponent. If he accelerates uniformly at \(4.0 \mathrm{~m} / \mathrm{s}^{2}\), (a) how long does it take him to catch his opponent, and (b) how far has he traveled in that time? (Assume the player with the puck remains in motion at constant speed.)

In 1865 Jules Verne proposed sending men to the Moon by firing a space capsule from a 220 -m-long cannon with final speed of \(10.97 \mathrm{~km} / \mathrm{s}\). What would have been the unrealistically large acceleration experienced by the space travelers during their launch? (A human can stand an acceleration of \(15 \mathrm{~g}\) for a short time.) Compare your answer with the free-fall acceleration, \(9.80 \mathrm{~m} / \mathrm{s}^{2}\).
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