2.A hiker used a compass to orient himself as he walked 3.00 km in a direction 50.0° north of west, and then an additional 2.00 km in a direction 20.0° south of west. The trip took 1.25 hours. Calculate the magnitude and direction of the hiker’s velocity of the hiker using vector components .
To calculate the magnitude and direction of the hiker's velocity using vector components, we need to find the x and y components of the displacement for each leg of the trip.
First, let's calculate the x and y components for the 3.00 km leg:
x = 3.00 km * cos(50°) ≈ 1.92 km
y = 3.00 km * sin(50°) ≈ 2.30 km
Next, let's calculate the x and y components for the 2.00 km leg:
x = 2.00 km * cos(160°) ≈ -1.98 km
y = 2.00 km * sin(160°) ≈ -0.342 km
Now, let's calculate the total x and y components by adding the respective components for both legs:
Total x-component = 1.92 km + (-1.98 km) ≈ -0.06 km
Total y-component = 2.30 km + (-0.342 km) ≈ 1.96 km
To find the magnitude of the hiker's velocity, we can use the formula:
Magnitude = sqrt(x^2 + y^2)
Magnitude = sqrt((-0.06 km)^2 + (1.96 km)^2) ≈ 1.96 km
To find the direction of the hiker's velocity, we can use the formula:
Direction = atan(y/x)
Direction = atan(1.96 km / -0.06 km) ≈ -87.9°
Therefore, the magnitude of the hiker's velocity is approximately 1.96 km and the direction is approximately 87.9° west of south.
To calculate the magnitude and direction of the hiker's velocity using vector components, we need to break down the hiker's displacement into its x and y components.
First, let's define the positive x-axis as east and the positive y-axis as north. We can calculate the x and y components of the hiker's displacement using trigonometry.
For the first leg of the trip (3.00 km in a direction 50.0° north of west), we need to find the x-component and y-component.
The x-component can be found by using the cosine of the angle:
x1 = 3.00 km * cos(50.0°)
The y-component can be found by using the sine of the angle:
y1 = 3.00 km * sin(50.0°)
Similarly, for the second leg of the trip (2.00 km in a direction 20.0° south of west), we can find the x and y components:
The x-component:
x2 = 2.00 km * cos(-20.0°)
(Note: Since the direction is south of west, the angle is negative.)
The y-component:
y2 = 2.00 km * sin(-20.0°)
Next, we need to find the total x and y components by summing up the components for each leg of the trip:
x_total = x1 + x2
y_total = y1 + y2
Now, we can calculate the magnitude of the hiker's velocity using the Pythagorean theorem:
magnitude = sqrt(x_total^2 + y_total^2)
Finally, we can calculate the direction of the hiker's velocity using trigonometry:
direction = atan2(y_total, x_total)
The resulting magnitude and direction will give us the hiker's velocity.
well, his final position in relation to his starting point is
(-3cos50°-2cos20°,3sin50°-2sin20°) = (-3.808,1.614)
and as usual, velocity = distance/time