A car travels 20.2 km due north and then
31.3 km in a direction φ = 61.2◦ west of north.
Find the magnitude of the car’s resultant
displacement.
Then Calculate the direction of the car’s resultant displacement, measured counterclockwise from the northerly direction.
Well, you know what they say about cars traveling in a westerly direction, right? They're just trying to get to their favorite "driving into the sunset" movie scene! But let's calculate their displacement first before we get caught up in the romantic vibes.
To find the car's resultant displacement, we can use the Pythagorean theorem because the displacement is essentially the hypotenuse of a right triangle formed by the car's northward and westward movements.
So, let's do the math: sqrt((20.2 km)^2 + (31.3 km)^2). And voila! The magnitude of the car's resultant displacement is approximately 37.1 km. Quite the scenic route, wouldn't you say?
Now, onto the direction. To find it, we can use trigonometry. The angle phi (φ) given is the angle west of north. So, simply subtract 61.2° from 90° (since north is perpendicular to the ground) to get the angle measured counterclockwise from the northerly direction.
Therefore, the direction of the car's resultant displacement is approximately 28.8° counterclockwise from the northerly direction. Just don't drive too far into the sunset now!
To find the magnitude of the car's resultant displacement, we can use the Pythagorean theorem.
The car first travels 20.2 km due north, which contributes to the displacement in the northerly direction.
Next, the car travels 31.3 km in a direction φ = 61.2° west of north. This contributes to both the northerly and westerly displacement components.
To calculate the westerly displacement component, we need to find the horizontal component of the 31.3 km distance. This can be done by multiplying the distance by the cosine of the angle φ. So, the westerly displacement component is 31.3 km multiplied by cosine (61.2°).
To calculate the northerly displacement component, we need to find the vertical component of the 31.3 km distance. This can be done by multiplying the distance by the sine of the angle φ. So, the northerly displacement component is 31.3 km multiplied by sine (61.2°).
Now, we can calculate the magnitude of the resultant displacement using the Pythagorean theorem:
Resultant displacement = √(northerly displacement)^2 + (westerly displacement)^2
Let's plug in the values:
Northerly displacement = 20.2 km
Westerly displacement = 31.3 km * cosine(61.2°)
Resultant displacement = √((20.2 km)^2 + (31.3 km * cosine(61.2°))^2)
Now, let's calculate the value.
To find the magnitude of the car's resultant displacement, we can use the Pythagorean theorem.
1. Draw a diagram representing the car's motion:
- Start at a point and go 20.2 km due north.
- From that point, go an additional 31.3 km at an angle of 61.2° west of north.
2. Calculate the horizontal and vertical components of displacement:
- The horizontal component is 31.3 km * sin(61.2°) = 26.99 km.
- The vertical component is 20.2 km + 31.3 km * cos(61.2°) = 56.04 km.
3. Use the Pythagorean theorem to calculate the magnitude of the resultant displacement:
- Magnitude = √(horizontal component^2 + vertical component^2)
- Magnitude = √((26.99 km)^2 + (56.04 km)^2)
- Magnitude ≈ √(727.06 km^2 + 3140.32 km^2)
- Magnitude ≈ √3867.38 km^2
- Magnitude ≈ 62.17 km.
Therefore, the magnitude of the car's resultant displacement is approximately 62.17 km.
To calculate the direction of the car's resultant displacement, you can use trigonometry.
4. Calculate the angle using:
- Angle = atan2(vertical component, horizontal component)
- Angle = atan2(56.04 km, 26.99 km)
- Angle ≈ 63.7°.
The direction of the car's resultant displacement, measured counterclockwise from the northerly direction, is approximately 63.7°.
α=180-61.2=118.8°
Cosine Law
d=sqrt{20.2²+31.3²-2•20.2•31.3•cos118.8°} =
=44,7 km
Sine Law
31.3/sinθ=44.7/sin118.8°
sinθ=31.3 •sin118.8°/44.7=0.61
θ= 37.9°