You’re riding your bicycle first west a distance of 7.7 km, then south a distance of 14.1 km.
(a)
In what direction does your average velocity vector point, in degrees south of west?
(b)
If the total trip took 18.3 minutes, what is the magnitude of this average velocity, in m/s?
(a) Well, since you're riding west and then south, we can say that your average velocity vector points southwest. But don't worry, I promise it's not as intimidating as it sounds. It's just a fancy way of saying that it's a bit diagonal, going in a direction that's kind of between west and south. So let's call it "west-southwest" and give ourselves a pat on the back for navigating such awesomely complicated directions!
(b) Now, let's calculate the magnitude of this average velocity in m/s. First, let's convert the total distance of 7.7 km + 14.1 km = 21.8 km to meters. Since 1 km = 1000 m, that gives us a total distance of 21,800 m.
Next, we need to convert the total trip time of 18.3 minutes to seconds. Since 1 minute = 60 seconds, we get a total time of 18.3 minutes * 60 seconds/minute = 1098 seconds.
Finally, we can calculate the magnitude of average velocity by dividing the total distance by the total time: 21,800 m / 1098 s = 19.85 m/s (rounded to two decimal places).
So, the magnitude of the average velocity is approximately 19.85 m/s. That's some super-speedy biking! Keep up the great work!
To find the direction of the average velocity vector, we can use the trigonometric functions to calculate the angle south of west.
(a)
First, we need to find the total displacement vector by adding the individual displacements in the x and y directions.
The displacement in the x direction (west) is 7.7 km.
The displacement in the y direction (south) is -14.1 km because it points in the opposite direction.
So the total displacement vector is (7.7 km, -14.1 km).
To find the angle, we can use the inverse tangent function (arctan):
Angle = arctan(-14.1 km / 7.7 km)
Using a calculator, we find that the angle is approximately -61.6 degrees.
Therefore, the average velocity vector points approximately 61.6 degrees south of west.
(b)
To find the magnitude of the average velocity, we need to convert the time from minutes to seconds because velocity is measured in m/s.
Given that the total trip took 18.3 minutes, we can convert it to seconds:
18.3 minutes * 60 seconds/minute = 1098 seconds.
Then, we can calculate the magnitude of the average velocity:
Magnitude of average velocity = Total displacement / Total time
= sqrt((7.7 km)^2 + (-14.1 km)^2) / 1098 seconds
Using a calculator, we find that the magnitude of the average velocity is approximately 0.0187 km/s.
However, since the final answer is requested in m/s, we need to convert kilometers to meters:
Magnitude of average velocity = 0.0187 km/s * 1000 m/km
= 18.7 m/s
Therefore, the magnitude of the average velocity is 18.7 m/s.
To answer both parts of the question, we need to calculate the average velocity vector.
(a) To find the direction of the average velocity vector, we can use trigonometry. The distance traveled west is 7.7 km and the distance traveled south is 14.1 km. If we draw a right triangle with these distances as the sides, the angle between the west direction and the average velocity vector will be the angle we are looking for.
Using the trigonometric function tangent, we can find this angle:
tan(angle) = (opposite/adjacent)
tan(angle) = (14.1 km / 7.7 km)
To find the angle itself, we can take the inverse tangent (also known as arctan) of both sides:
angle = arctan(14.1 km / 7.7 km)
Calculating this using a calculator gives us the angle in radians. To convert to degrees, we can multiply by (180/π) where π is approximately 3.14159.
angle (in degrees) = arctan(14.1 km / 7.7 km) * (180/π)
(b) To find the magnitude of the average velocity, we divide the total distance traveled by the total time taken. The total distance is the square root of the sum of the squares of the distances traveled in each direction:
total distance = sqrt((7.7 km)^2 + (14.1 km)^2)
To convert the total trip time from minutes to seconds, we multiply by 60:
total time = 18.3 minutes * 60 seconds/minute
The magnitude of the average velocity is then:
magnitude = total distance / total time
Now, let's calculate the answers.
(a) Use the formula tan(angle) = (14.1 km / 7.7 km) to find the angle in radians. Then convert it to degrees.
(b) Calculate the total distance using the formula total distance = sqrt((7.7 km)^2 + (14.1 km)^2). Then calculate the total time in seconds. Finally, divide the total distance by the total time to find the magnitude of the average velocity.
Please note that for more accurate calculations, you may need to consider significant figures and unit conversions.
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