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Chapter 3: Problem 73

At an antique car rally, a Stanley Steamer automobile travels north at $40 \mathrm{km} / \mathrm{h}\( and a Pierce Arrow automobile travels east at \)50 \mathrm{km} / \mathrm{h}$. Relative to an observer riding in the Stanley Steamer, what are the \(x\) - and \(y\) -components of the velocity of the Pierce Arrow car? The \(x\) -axis is to the east and the \(y\) -axis is to the north.

Short Answer

Expert verified
Answer: The x-component of the relative velocity is 50 km/h (to the east) and the y-component of the relative velocity is -40 km/h (to the south).

Step by step solution

01

Understand and organize given information

We are given the velocities of both cars: - Stanley Steamer velocity: 40 km/h to the north (\(v_{SS}\)) - Pierce Arrow velocity: 50 km/h to the east (\(v_{PA}\)) The x-axis is to the east and the y-axis is to the north. We need to find the velocity components of the Pierce Arrow car relative to the Stanley Steamer car (\(v_{PArel}\)) in the x- and y-axis.
02

Define the relative velocity of Pierce Arrow car

We can define the relative velocity by subtracting the velocity of Stanley Steamer from that of Pierce Arrow: \(v_{PArel} = v_{PA} - v_{SS}\) Since their directions are perpendicular to each other, the x-component of velocity of the Pierce Arrow car will not be affected by the Stanley Steamer car's velocity, and the y-component of the velocity of the Pierce Arrow car will change by the amount of the negative Stanley Steamer car's velocity.
03

Calculate the x-component of the relative velocity

As mentioned in step 2, the x-component of the velocity of the Pierce Arrow car will not be affected by the Stanley Steamer's velocity: \(v_{PArel_x} = v_{PA_x}\) As the Pierce Arrow car is traveling east at 50 km/h: \(v_{PArel_x} = 50\,\mathrm{km/h}\)
04

Calculate the y-component of the relative velocity

The y-component of the Pierce Arrow car relative to Stanley Steamer car will change by the amount of the negative Stanley Steamer car's velocity: \(v_{PArel_y} = v_{PA_y} - v_{SS_y}\) Since the Pierce Arrow car is traveling horizontally and not in the north direction, its y-component is 0: \(v_{PA_y} = 0\,\mathrm{km/h}\) The Stanley Steamer car is traveling north, so its y-component should be subtracted from the Pierce Arrow car's y-component: \(v_{PArel_y} = 0\,\mathrm{km/h} - 40\,\mathrm{km/h} = -40\,\mathrm{km/h}\)
05

Summarize the results

The relative velocity components of the Pierce Arrow car while observing from the Stanley Steamer car are: - x-component: \(v_{PArel_x} = 50\,\mathrm{km/h}\) (to the east) - y-component: \(v_{PArel_y} = -40\,\mathrm{km/h}\) (to the south)

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