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Chapter 3: Problem 34

A car travels east at \(96 \mathrm{km} / \mathrm{h}\) for \(1.0 \mathrm{h}\). It then travels \(30.0^{\circ}\) east of north at \(128 \mathrm{km} / \mathrm{h}\) for \(1.0 \mathrm{h} .\) (a) What is the average speed for the trip? (b) What is the average velocity for the trip?

Short Answer

Expert verified
Answer: The average speed for the trip is 112 km/h, and the average velocity for the trip is 108.05 km/h.

Step by step solution

01

Determine the distance in each direction

We can find the distance traveled in each direction by using the formula for distance: distance = speed × time taken. For the first part of the trip, the car travels east at 96 km/h for 1 hour, and for the second part, it goes 30 degrees east of north at 128 km/h for 1 hour. First part: Distance_east = 96 km/h × 1 h = 96 km Second part: Distance_30degrees = 128 km/h × 1 h = 128 km
02

Find the total distance traveled

The total distance is the sum of individual distances taken in each direction. Total distance = Distance_east + Distance_30degrees = 96 km + 128 km = 224 km
03

Calculate the average speed for the trip

The average speed is defined as the total distance traveled divided by the total time taken. In this case, the total time taken for the trip is 2.0 hours (1 hour for each leg of the trip). Average speed = Total distance / Total time = 224 km / 2.0 h = 112 km/h
04

Determine the displacement in each direction

To find the displacement for each direction, we break the second part of the trip (going 30 degrees east from north) into its vertical (north) and horizontal (east) components. We can use sine and cosine functions to calculate these components: Displacement_east = Distance_east + Distance_30degrees_cos(30) Displacement_north = Distance_30degrees_sin(30) Displacement_east = 96 km + 128 km*cos(30) = 96 km + 128 km*(0.866) = 96 km + 110.9 km = 206.9 km Displacement_north = 128 km*sin(30) = 128 km*(0.5) = 64 km
05

Calculate the total displacement

Using the Pythagorean theorem, we can find the total displacement of the car: Total_displacement = sqrt(Displacement_east^2 + Displacement_north^2) Total_displacement = sqrt((206.9 km)^2 + (64 km)^2) = sqrt(46690.41 km^2) = 216.1 km
06

Calculate the average velocity for the trip

Average velocity is defined as the total displacement divided by the total time taken. As mentioned before, the total time taken is 2.0 hours. Average velocity = Total displacement / Total time = 216.1 km / 2.0 h = 108.05 km/h Now we have the answers for both parts: (a) The average speed for the trip: 112 km/h (b) The average velocity for the trip: 108.05 km/h

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