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Chapter 2: Problem 58

A speedboat moving at \(30.0 \mathrm{~m} / \mathrm{s}\) approaches a no-wake buoy marker \(100 \mathrm{~m}\) ahead. The pilot slows the boat with a constant acceleration of \(-3.50 \mathrm{~m} / \mathrm{s}^{2}\) by reducing the throttle. (a) How long does it take the boat to reach the buoy? (b) What is the velocity of the boat when it reaches the buoy?

Short Answer

Expert verified
(a) It takes the boat approximately 5.612 seconds to reach the buoy. (b) The velocity of the boat when it reaches the buoy is approximately 10.358 m/s.

Step by step solution

01

Identify the knowns and unknowns.

The knowns are: initial speed of the boat \(v_{i} = 30.0 ~m/s\), distance to the buoy \(d = 100 ~m\) and acceleration \(a = -3.50 ~m/s^2\). The unknowns are the time \(t\) and the final velocity \(v_{f}\).
02

Use the equation of motion to calculate the time.

By using the first equation of motion \(d = v_{i}t + 0.5at^2\), we have \(100 = 30t - 1.75t^2\). Now we solve this quadratic equation.
03

Calculate the roots of the equation.

The solutions of the quadratic equation are given by the formula \(t=(-b \pm \sqrt{b^2-4ac})/(2a)\), where \(a = -1.75, b = 30\) and \(c = -100\). Solving, we find the roots \(t1 = 5.612 ~s\) and \(t2 = 10.210 ~s\). Since time can't be negative, we discard the larger root. Thus, \(t = t1 = 5.612 ~s\).
04

Use the equation of motion to calculate the velocity.

By using the second equation of motion \(v = v_i + at\), we have \(v_f = 30 - 3.50*5.612\). Solving, we find \(v_f = 10.358 ~m/s\).

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Most popular questions from this chapter

BIO Colonel John P. Stapp, USAF, participated in studying whether a jet pilot could survive emergency ejection. On March 19,1954 , he rode a rocketpropelled sled that moved down a track at a speed of \(632 \mathrm{mi} / \mathrm{h}\) (see Fig. P2.56). He and the sled were safely brought to rest in \(1.40 \mathrm{~s}\). Determine in SI units (a) the negative acceleration he experienced and (b) the distance he traveled during this negative acceleration.

An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at \(+40 \mathrm{ft} / \mathrm{s}^{2}\). After some time \(t_{1}\), the rocket engine is shut down and the sled moves with constant velocity \(v\) for a time \(t_{2}\). If the total distance traveled by the sled is \(17500 \mathrm{ft}\) and the total time is \(90 \mathrm{~s}\), find (a) the times \(t_{1}\) and \(t_{2}\) and (b) the velocity \(v\). At the \(17500-\mathrm{ft}\) mark, the sled begins to accelerate at \(-20 \mathrm{ft} / \mathrm{s}^{2}\). (c) What is the final position of the sled when it comes to rest? (d) How long does it take to come to rest?

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.)

A person travels by car from one city to another with different constant speeds between pairs of cities. She drives for \(30.0 \mathrm{~min}\) at \(80.0 \mathrm{~km} / \mathrm{h}, 12.0 \mathrm{~min}\) at \(100 \mathrm{~km} / \mathrm{h}\), and \(45.0 \mathrm{~min}\) at \(40.0 \mathrm{~km} / \mathrm{h}\) and spends \(15.0 \mathrm{~min}\) eating lunch and buying gas. (a) Determine the average speed for the trip. (b) Determine the distance between the initial and final cities along the route.

A package is dropped from a helicopter that is descending steadily at a speed \(v_{0}\). After \(t\) seconds have elapsed, (a) what is the speed of the package in terms of \(v_{0}, g\), and \(t\) ? (b) What distance \(d\) is it from the helicopter in terms of \(g\) and \(t\) ? (c) What are the answers to parts (a) and (b) if the helicopter is rising steadily at the same speed?
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