Peggy drives from Cornwall to Atkins Glen in 37 min. Cornwall is 74.8 km from Illium in a direction 22¡Æ west of south. Atkins Glen is 24.6 km from Illium in a direction 18¡Æ south of west. Use Illium as your origin.
Find the displacement during the trip.
magnitude ___ km
direction ___¡Ænorth of east
Find Peggy's average velocity for the trip.
magnitude ___ km/h
direction ___¡Ænorth of east
i don't know about the answer, i need a solution for this question
To find the displacement during Peggy's trip, we need to find the net change in position from Illium to Atkins Glen.
First, let's break down the motion into its components:
- Cornwall is 74.8 km from Illium in a direction 22° west of south.
- Atkins Glen is 24.6 km from Illium in a direction 18° south of west.
To find the displacement, we need to add the displacement vectors for each leg of the trip.
Step 1: Convert the directions to angles east of north:
- Cornwall: 180° - 22° = 158° east of north.
- Atkins Glen: 270° - 18° = 252° east of north.
Step 2: Convert the angles to radians:
- Cornwall: 158° * (π/180) ≈ 2.758 radians.
- Atkins Glen: 252° * (π/180) ≈ 4.403 radians.
Step 3: Calculate the x and y components for each leg of the trip:
- Cornwall: x = 74.8 km * cos(2.758) ≈ 69.865 km; y = 74.8 km * sin(2.758) ≈ -8.141 km.
- Atkins Glen: x = 24.6 km * cos(4.403) ≈ -3.882 km; y = 24.6 km * sin(4.403) ≈ -24.034 km.
Step 4: Calculate the net change in position by summing the x and y components:
- Δx = 69.865 km - 3.882 km ≈ 65.983 km.
- Δy = -8.141 km - (-24.034 km) ≈ 15.893 km.
The displacement is given by the net change in position, which is (√(Δx^2 + Δy^2)) km:
√(65.983^2 + 15.893^2) ≈ √4483.349 ≈ 66.98 km.
Therefore, the magnitude of the displacement is approximately 66.98 km.
The direction of the displacement is given by the inverse tangent of Δy/Δx:
θ = tan^(-1)(15.893/65.983) ≈ tan^(-1)(0.240) ≈ 13.774°.
Since the displacement is north of east, the direction is 90° - 13.774° ≈ 76.226° north of east.
Therefore, the direction of the displacement is approximately 76.226° north of east.
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To find Peggy's average velocity for the trip, we need to divide the displacement by the time it took.
Given:
- Displacement: 66.98 km.
- Time: 37 min.
Step 1: Convert the time to hours:
- Time: 37 min ÷ 60 min/hour ≈ 0.617 hours.
Step 2: Calculate the average velocity by dividing the displacement by the time:
- Average Velocity = 66.98 km / 0.617 hours ≈ 108.536 km/h.
Therefore, the magnitude of Peggy's average velocity for the trip is approximately 108.536 km/h.
The direction of Peggy's average velocity is the same as the direction of the displacement since it is a vector quantity.
Therefore, the direction of Peggy's average velocity is approximately 76.226° north of east.