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Chapter 3: Problem 87

A particle has a constant acceleration of \(5.0 \mathrm{m} / \mathrm{s}^{2}\) to the east. At time \(t=0,\) it is \(2.0 \mathrm{m}\) east of the origin and its velocity is \(20 \mathrm{m} / \mathrm{s}\) north. What are the components of its position vector at \(t=2.0 \mathrm{s} ?\)

Short Answer

Expert verified
Answer: The components of the position vector at t=2 seconds are 12.0 m eastward and 40.0 m northward.

Step by step solution

01

Determine the eastward component of position

Since the particle's motion in the eastward direction is under constant acceleration, we can use the equation: \(x(t) = x_0 + v_{0x}t + \frac{1}{2}at^2\). Here, \(x_0 = 2.0 m\), \(v_{0x} = 0\) (Because the initial velocity is in the northward direction, there is no eastward component of initial velocity), and \(a = 5.0 \mathrm{m} / \mathrm{s}^2\). Now we can plug in these values and given time, \(t=2s\), into the equation to get the eastward component of the position: \(x(2) = 2.0 + 0(2) + \frac{1}{2}(5)(2)^2 = 2.0 + 10 = 12.0 \mathrm{m}\) eastward.
02

Determine the northward component of position

In the northward direction, since there is no acceleration, the particle moves with constant velocity. We can use the equation: \(y(t) = y_0 + v_{0y}t\) Here, \(y_0 = 0\) (particle starts from the origin, so there is no initial position in the northward direction) and \(v_{0y} = 20 \mathrm{m} / \mathrm{s}\). Now we can plug in these values and given time, \(t=2s\), into the equation to get the northward component of the position: \(y(2) = 0 + 20(2) = 40 \mathrm{m}\) northward.
03

Write the position vector components

After calculating both eastward and northward components of position, we have the position vector at \(t=2.0s\): \((\)12.0 m eastward, 40.0 m northward\()\).

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Most popular questions from this chapter

Jerry bicycles from his dorm to the local fitness center: 3.00 miles east and 2.00 miles north. Cindy's apartment is located 1.50 miles west of Jerry's dorm. If Cindy is able to meet Jerry at the fitness center by bicycling in a straight line, what is the length and direction she must travel?
Geoffrey drives from his home town due east at \(90.0 \mathrm{km} / \mathrm{h}\) for 80.0 min. After visiting a friend for 15.0 min, he drives in a direction \(30.0^{\circ}\) south of west at \(76.0 \mathrm{km} / \mathrm{h}\) for 45.0 min to visit another friend. (a) How far is it to his home from the second town? (b) If it takes him 45.0 min to drive directly home, what is his average velocity on the third leg of the trip? (c) What is his average velocity during the first two legs of his trip? (d) What is his average velocity over the entire trip? (e) What is his average speed during the entire trip if he spent 55.0 min visiting the second friend?
Two cars are driving toward each other on a straight, flat Kansas road. The Jeep Wrangler is traveling at \(82 \mathrm{km} / \mathrm{h}\) north and the Ford Taurus is traveling at \(48 \mathrm{km} / \mathrm{h}\) south, both measured relative to the road. What is the velocity of the Jeep relative to an observer in the Ford?
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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.
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