Anonymous
Anonymous asked in Science & MathematicsMathematics · 5 months ago

Help? Vector math problem??

A car travels east at 100 km/h for 3 h, and then N45E at 80 km/h for 2 h.

a) Determine the magnitude of the car's displacement.

b) Determine the bearing of the car.

2 Answers

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  • 5 months ago
    Best answer

    Recall: s = d/t → where s is the speed, d is the distance, t is the time

    First step: the car travels EAST at 100 km/h for 3 h

    s₁ = d₁/t₁

    d₁ = s₁.t₁ → given that: s₁ = 100 km/h

    d₁ = 100.t₁ → given that: t₁ = 3 h

    d₁ = 300 km

    Second step: the car travels N45E at 80 km/h for 2 h

    s₂ = d₂/t₂

    d₂ = s₂.t₂ → given that: s₂ = 80 km/h

    d₂ = 80.t₂ → given that: t₂ = 2 hours

    d₂ = 160 km

    By using the Pythagorean's theorem, you can see that:

    OC² = OB² + BC²

    OC² = [OA + AB]² + BC²

    OC² = [d₁ + d₂.cos(45)]² + [d₂.sin(45)]²

    OC² = d₁² + 2.d₁.d₂.cos(45) + d₂².cos²(45) + d₂².sin²(45)

    OC² = d₁² + 2.d₁.d₂.cos(45) + d₂².[cos²(45) + sin²(45)] → recall: cos²(x) + sin²(x) = 1

    OC² = d₁² + 2.d₁.d₂.cos(45) + d₂²

    OC² = 300² + (2 * 300 * 160).[(√2)/2] + 160²

    OC² = 90000 + 48000√2 + 25600

    OC² = 115600 + 48000√2

    OC² = 400.(289 + 120√2)

    OC = √[400.(289 + 120√2)]

    OC = 20√(289 + 120√2)

    OC ≈ 428.35 km

    Recall:

    s = d/t

    s = (d₁ + d₂) / (t₁ + t₂)

    s = (300 + 160) / (3 + 2)

    s = 460/5

    s = 92 km/h

    Attachment image
  • TomV
    Lv 7
    5 months ago

    a) Use the law of cosines to calculate the displacement

    Distance traveled East = 100*3 = 300 km

    Distance traveled NorthEast = 80*2 = 160 km

    If the displacement of the car, c, is the base of a triangle, the legs are 160 km and 300 km and the apex angle of the triangle is 90 + 45 = 135°

    c² = 300² + 160² - 2(300)(160)cos(135°)

    Ans: c = 428.3 km

    b) Use the law of sines to calculate the displacement angle relative to E

    428.3/sin135° = 160/sinΘ

    sinΘ = (160/428.3)sin(135) = 0.26415

    Θ = arcsin(0.26415) = 15.3° North of East or 74.7° East of North

    Ans: N74.7E

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