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 27m/s, and the average southward velocity has a magnitude of 17m/s. What is the average velocity, magnitude and direction of the displacem,ent vector
Well, it seems like this car is having quite the journey! Let's calculate the average velocity and displacement using a bit of mathematical clownery.
First, let's assume the total time taken for the trip is T seconds. Since the car travels in opposite directions during the trip, we can say that the time spent going north is 3T/4 seconds, and the time spent going south is T/4 seconds.
Now, let's calculate the total displacement during the northward journey. We can use the formula v = d/t, where v is the velocity and d is the displacement. Rearranging the formula, we have d = v * t. So, the displacement during the northward journey is (27 m/s) * (3T/4 s) = 81T/4 meters.
Similarly, the displacement during the southward journey is (17 m/s) * (T/4 s) = 17T/4 meters.
To find the total displacement, we can add the northward and southward displacements together: 81T/4 + 17T/4 = 98T/4 = 24.5T meters.
Now, let's find the total time for the entire trip. It's just the sum of the times spent going north and south, so T + T/4 = 5T/4 seconds.
Finally, we can calculate the average velocity by dividing the total displacement by the total time: (24.5T meters) / (5T/4 seconds) = 24.5 * 4 / 5 = 19.6 m/s.
Therefore, the average velocity (magnitude) of the displacement vector is approximately 19.6 m/s. As for the direction, it should be noted that the displacement is predominantly due north, so we can say the average displacement vector points in the north direction.
Hope that answers your question! Safe travels, my friend!
To find the average velocity and direction of the displacement vector, we need to calculate the resultant velocity by considering both the northward and southward velocities.
Let's assume the total time for the trip is "T" seconds.
Given:
- The car travels due north for three-fourths of the time, which means it travels for (3/4)T seconds.
- The northward velocity magnitude is 27 m/s.
- The car travels due south for one-fourth of the time, which means it travels for (1/4)T seconds.
- The southward velocity magnitude is 17 m/s.
To find the average velocity, we need to calculate the resultant velocity vector.
Step 1: Calculate the displacement vector for the northward motion.
The displacement vector for the northward motion is given by:
Displacement_north = (northward velocity magnitude) * (time taken for northward motion)
Displacement_north = 27 m/s * (3/4)T
Step 2: Calculate the displacement vector for the southward motion.
The displacement vector for the southward motion is given by:
Displacement_south = (southward velocity magnitude) * (time taken for southward motion)
Displacement_south = -17 m/s * (1/4)T
(Negative sign indicates motion in the opposite direction to the positive axis)
Step 3: Calculate the resultant displacement vector.
Resultant displacement vector = Displacement_north + Displacement_south
Resultant displacement vector = 27 m/s * (3/4)T - 17 m/s * (1/4)T
= 20.25 m/s * T - 4.25 m/s * T
= (20.25 - 4.25) m/s * T
= 16 m/s * T
The magnitude of the resultant displacement vector is 16 m/s, and its direction is due north.
Therefore, the average velocity of the displacement vector is 16 m/s due north.
To find the average velocity of the displacement vector, we need to first calculate the total distance traveled in each direction.
Let's assume that the total time taken for the trip is t. Thus, the car travels for 3/4t in the north direction and for 1/4t in the south direction.
Distance traveled in the north direction = (3/4t) * (27m/s) = (81/4)t m
Distance traveled in the south direction = (1/4t) * (17m/s) = (17/4)t m
Now, to calculate the total displacement, we need to take into account both the magnitude and direction.
Since the car travels due north and due south, the displacement in the north direction is positive and the displacement in the south direction is negative.
Displacement in the north direction = (81/4)t m
Displacement in the south direction = (-17/4)t m
To find the average displacement, we need to add the displacements in both directions:
Average displacement = (81/4 - 17/4)t m
Average displacement = (64/4)t m
Average displacement = (16t) m
Therefore, the average displacement magnitude is 16t m.
To determine the direction of the average displacement, we look at whether the displacement is positive or negative. In this case, since the displacement in the south direction is negative, the average displacement points south.
Hence, the average displacement magnitude is 16t m in the south direction.
Assume t= 1 minute = 60 s.
From v = d/ t, equation for d will be v.t
Individually solve for d for north & south, you'll arrive at +1215 m for north displacement, and -255 m for south.
Add these values to get total displacement. Answer for d sub t = +960 m.
Thus the magnitude of displacement vector is 960 m, north direction (as indicated by the + sign).
Solving for average velocity, use total displacement divided by the total elapsed time:
+960m / 60 s = 16 m/s, direction is north (positive answer)