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Chapter 3: Problem 33

A bicycle travels \(3.2 \mathrm{km}\) due east in \(0.10 \mathrm{h}\), then $4.8 \mathrm{km}\( at \)15.0^{\circ}\( east of north in \)0.15 \mathrm{h},$ and finally another \(3.2 \mathrm{km}\) due east in \(0.10 \mathrm{h}\) to reach its destination. The time lost in turning is negligible. What is the average velocity for the entire trip?

Short Answer

Expert verified
Answer: The average velocity for the entire trip is approximately 31.863 km/h.

Step by step solution

01

Calculate the displacement for each part of the trip

To calculate the displacement of each part of the trip, we need to find the horizontal (x) and vertical (y) components of the displacement for each part, and then add them up. For the parts where the bicycle is moving due east, the entire displacement is in the x-direction. When moving at an angle, we can use trigonometry (sine and cosine functions) to find the x and y components of the displacement.
02

Calculate horizontal and vertical displacement components for the second part of the trip

For the second part of the trip, the bicycle is traveling 4.8 km at an angle of 15 degrees east of north. To find the x and y components of this displacement, we can use the sine and cosine functions: x = 4.8 * cos(15) = 4.8 * 0.9659 = 4.6365 km y = 4.8 * sin(15) = 4.8 * 0.2588 = 1.2422 km
03

Calculate the total displacement in the x and y directions

To find the total displacement, we can simply add up the displacements in the x and y directions for each part of the trip: Total x displacement = 3.2 + 4.6365 + 3.2 = 11.0365 km Total y displacement = 0 + 1.2422 + 0 = 1.2422 km
04

Calculate the magnitude of the total displacement

To find the magnitude of the total displacement, we can use the Pythagorean theorem: Total displacement = sqrt((total x displacement)^2 + (total y displacement)^2) Total displacement = sqrt((11.0365)^2 + (1.2422)^2) = sqrt(122.8035 + 1.5435) = sqrt(124.3471) = 11.152 km
05

Calculate the total time taken for the trip

The total time taken for the trip is the sum of the times taken for each part of the trip: Total time = 0.10 h + 0.15 h + 0.10 h = 0.35 h
06

Calculate the average velocity for the entire trip

To find the average velocity, we can divide the total displacement by the total time taken: Average velocity = Total displacement / Total time = 11.152 km / 0.35 h = 31.863 km/h The average velocity for the entire trip is approximately 31.863 km/h.

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Most popular questions from this chapter

You want to make a plot of the trajectory of a projectile. That is, you want to make a plot of the height \(y\) of the projectile as a function of horizontal distance \(x\). The projectile is launched from the origin with an initial speed \(v_{\mathrm{i}}\) at an angle \(\theta\) above the horizontal. Show that the equation of the trajectory followed by the projectile is $$y=\left(\frac{v_{\text {iy }}}{v_{\text {ix }}}\right) x+\left(\frac{-g}{2 v_{\text {ix }}^{2}}\right) x^{2}$$
A speedboat heads west at \(108 \mathrm{km} / \mathrm{h}\) for 20.0 min. It then travels at \(60.0^{\circ}\) south of west at \(90.0 \mathrm{km} / \mathrm{h}\) for 10.0 min. (a) What is the average speed for the trip? (b) What is the average velocity for the trip?
A scout troop is practicing its orienteering skills with map and compass. First they walk due east for \(1.2 \mathrm{km}\) Next, they walk \(45^{\circ}\) west of north for \(2.7 \mathrm{km} .\) In what direction must they walk to go directly back to their starting point? How far will they have to walk? Use graph paper, ruler, and protractor to find a geometrical solution.
A gull is flying horizontally \(8.00 \mathrm{m}\) above the ground at $6.00 \mathrm{m} / \mathrm{s} .$ The bird is carrying a clam in its beak and plans to crack the clamshell by dropping it on some rocks below. Ignoring air resistance, (a) what is the horizontal distance to the rocks at the moment that the gull should let go of the clam? (b) With what speed relative to the rocks does the clam smash into the rocks? (c) With what speed relative to the gull does the clam smash into the rocks?
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