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It looks like you've already started sketching the vector diagram in your image, which is a great start! Your sketch correctly shows the three displacement vectors and the resultant total displacement

Physics

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Answer

b

81.08 km at 39.56∘ North of East81.08 \text{ km at } 39.56^\circ \text{ North of East}

Step-by-step solution

4 steps

It looks like you've already started sketching the vector diagram in your image, which is a great start! Your sketch correctly shows the three displacement vectors and the resultant total displacement.

Here's a detailed breakdown of the solution:

Part a

Sketch the vector diagram

Your sketch accurately represents the three displacement vectors:

  1. A vector D1\mathbf{D_1} of 50 km pointing East (along the positive x-axis).
  2. A vector D2\mathbf{D_2} of 30 km pointing North (along the positive y-axis), starting from the end of D1\mathbf{D_1}.
  3. A vector D3\mathbf{D_3} of 25 km at 30∘30^\circ East of North, starting from the end of D2\mathbf{D_2}. This means the angle is measured 30∘30^\circ from the North axis towards the East.
  4. The resultant total displacement vector, Dtotal\mathbf{D_{total}}, drawn from the origin (start of D1\mathbf{D_1}) to the end of D3\mathbf{D_3}.

Part b

81.08 km at 39.56∘ North of East81.08 \text{ km at } 39.56^\circ \text{ North of East}

Determine the total displacement of the car

  1. Resolve each displacement vector into its East (x) and North (y) components.

    Let East be the positive x-direction and North be the positive y-direction.

    For D1=50 km East\mathbf{D_1} = 50 \text{ km East}:

    D1x=50 kmD_{1x} = 50 \text{ km} D1y=0 kmD_{1y} = 0 \text{ km}

    For D2=30 km North\mathbf{D_2} = 30 \text{ km North}:

    D2x=0 kmD_{2x} = 0 \text{ km} D2y=30 kmD_{2y} = 30 \text{ km}

    For D3=25 km at 30∘ East of North\mathbf{D_3} = 25 \text{ km at } 30^\circ \text{ East of North}:
    The angle 30∘30^\circ is with respect to the North axis.

    D3x=25sin⁡(30∘)=25×0.5=12.5 kmD_{3x} = 25 \sin(30^\circ) = 25 \times 0.5 = 12.5 \text{ km} D3y=25cos⁡(30∘)=25×32≈25×0.8660=21.65 kmD_{3y} = 25 \cos(30^\circ) = 25 \times \frac{\sqrt{3}}{2} \approx 25 \times 0.8660 = 21.65 \text{ km}
  2. Sum the x-components and y-components to find the total displacement components.

    Total East displacement (DxD_x):

    Dx=D1x+D2x+D3x=50+0+12.5=62.5 kmD_x = D_{1x} + D_{2x} + D_{3x} = 50 + 0 + 12.5 = 62.5 \text{ km}

    Total North displacement (DyD_y):

    Dy=D1y+D2y+D3y=0+30+21.65=51.65 kmD_y = D_{1y} + D_{2y} + D_{3y} = 0 + 30 + 21.65 = 51.65 \text{ km}
  3. Calculate the magnitude of the total displacement.

    The magnitude DtotalD_{total} is found using the Pythagorean theorem:

    Dtotal=Dx2+Dy2D_{total} = \sqrt{D_x^2 + D_y^2} Dtotal=(62.5)2+(51.65)2D_{total} = \sqrt{(62.5)^2 + (51.65)^2} Dtotal=3906.25+2667.7225D_{total} = \sqrt{3906.25 + 2667.7225} Dtotal=6573.9725D_{total} = \sqrt{6573.9725} Dtotal≈81.08 kmD_{total} \approx 81.08 \text{ km}
  4. Calculate the direction of the total displacement.

    The direction θ\theta is found using the arctangent function, measured from the East axis towards the North:

    θ=arctan⁡(DyDx)\theta = \arctan\left(\frac{D_y}{D_x}\right) θ=arctan⁡(51.6562.5)\theta = \arctan\left(\frac{51.65}{62.5}\right) θ=arctan⁡(0.8264)\theta = \arctan(0.8264) θ≈39.56∘\theta \approx 39.56^\circ

    Since both DxD_x and DyD_y are positive, the direction is North of East.

    The total displacement of the car is 81.08 km at 39.56∘ North of East\boxed{81.08 \text{ km at } 39.56^\circ \text{ North of East}}.

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