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Chapter 2: Problem 1

Consider two displacements, one of magnitude \(3 \mathrm{~m}\) and another of magnitude \(4 \mathrm{~m}\). Show how the displacement vectors may be combined to get a resultant displacement of magnitude \((\) a) \(7 \mathrm{~m},(b) 1 \mathrm{~m}\), and \((c) 5 \mathrm{~m}\)

Short Answer

Expert verified
The vectors can be arranged in three different ways to get the desired results. For a resultant displacement of 7m, the vectors of 3m and 4m must be arranged in the same direction. For a resultant displacement of 1m, the vectors must be directed opposite each other. For a resultant displacement of 5m, the vectors should be arranged at right angles to each other.

Step by step solution

01

Solving for 7m magnitude resultant

Place the two vectors end to end in the same direction, starting from the same point. When two vectors are added together, the resulting vector, or the resultant, simply combines the magnitudes of the two original vectors. In this case, \(3 m + 4 m = 7 m\). Therefore, aligning the displacements in the same direction will result in a resultant displacement of \(7 m\).
02

Solving for 1m magnitude resultant

Proceed in a similar manner, but now place the two vectors in opposite directions, starting from the same point. With the vectors arranged in this manner, the magnitudes subtract from each other instead of add together, \(4 m - 3 m = 1 m\). Therefore, arranging the displacements opposite to one another will result in a resultant displacement of \(1 m\).
03

Solving for 5m magnitude resultant

Arranging two vectors at right angles to each other will add the squares of their magnitudes, the square root of the sum will be the magnitude of the resultant. Therefore the magnitude of the resultant, R, can be given by \(R = \sqrt{(3^2 + 4^2)} = 5 m\). Thus, a displacement of \(5m\) will be obtained when the given vectors are arranged at right angles to each other.

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Most popular questions from this chapter

A particle undergoes three successive displacements in a plane, as follows: \(4.13 \mathrm{~m}\) southwest, \(5.26 \mathrm{~m}\) east, and \(5.94 \mathrm{~m}\) in a direction \(64.0^{\circ}\) north of east. Choose the \(x\) axis pointing east and the \(y\) axis pointing north and find \((a)\) the components of each displacement, \((b)\) the components of the resultant displacement, \((c)\) the magnitude and direction of the resultant displacement, and \((d)\) the displacement that would be required to bring the particle back to the starting point.

A muon (an elementary particle) is shot with initial speed \(5.20 \times 10^{6} \mathrm{~m} / \mathrm{s}\) into a region where an electric field produces an acceleration of \(1.30 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2}\) directed opposite to the initial velocity. How far does the muon travel before coming to rest?

Raindrops fall to the ground from a cloud \(1700 \mathrm{~m}\) above Earth's surface. If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? Would it be safe to walk outside during a rainstorm?

A woman walks \(250 \mathrm{~m}\) in the direction \(35^{\circ}\) east of north, then \(170 \mathrm{~m}\) directly east. (a) Using graphical methods, find her final displacement from the starting point. (b) Compare the magnitude of her displacement with the distance she walked.

A car is driven east for a distance of \(54 \mathrm{~km}\), then north for \(32 \mathrm{~km}\), and then in a direction \(28^{\circ}\) east of north for \(27 \mathrm{~km}\). Draw the vector diagram and determine the total displacement of the car from its starting point.
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