A car makes a trip due north for three-fourths of the time and due south one-fourth of the time. The average northward velocity has a magnitude of 25 m/s, and the average southward velocity has a magnitude of 32 m/s. Taking northward to be the positive direction, what is the average velocity for the trip?
pick some times that meet the criterion
north for 30 seconds, south for 10 seconds , total time = 40 seconds
distance north from start = 25*30
then back 32*10
total distance north (displacement vector, not distance traveled) = 25*30 - 32*10
so
Vaverage = (25*30 - 32*10)/40
= (75 - 32)/4
= 10.75 m/s
To find the average velocity for the trip, we need to calculate the average of the northward and southward velocities.
First, let's find the magnitude of the average velocity for the northward motion.
Given:
Northward average velocity = 25 m/s
Time spent traveling north = 3/4
Average velocity for the northward motion = (northward average velocity) * (time spent traveling north)
Average velocity for the northward motion = 25 m/s * 3/4 = 18.75 m/s
Similarly, let's find the magnitude of the average velocity for the southward motion.
Given:
Southward average velocity = 32 m/s
Time spent traveling south = 1/4
Average velocity for the southward motion = (southward average velocity) * (time spent traveling south)
Average velocity for the southward motion = 32 m/s * 1/4 = 8 m/s
Now, to find the average velocity for the entire trip, we need to consider the direction of motion.
Given that northward is considered the positive direction, and the car travels both northward and southward, the average velocity for the trip will have a positive value.
Since the magnitude of the northward average velocity (18.75 m/s) is greater than the magnitude of the southward average velocity (8 m/s), the average velocity for the trip will have the same direction as the northward velocity.
Therefore, the average velocity for the trip is 18.75 m/s in the northward direction.