Respuesta :
Answer: The car's displacement has magnitude 228.3381 ft and direction of 7.46°.
Explanation: Displacement is defined as the change in position of an object. It is a vector, so it has magnitude and direction.
To determine the car's displacement, divide its route into 3 parts:
1) Horizontally for 202ft;
2) Climbs 147ft at an angle of 32°;
3) Moves 147ft at an angle of 47° below the horizontal;
Now calculate each x- and y- displacement of each part.
Part (1):
The car only displaces on x-axis, so displacement is 202ft.
Part (2):
The car climbs at an angle with the horizontal, so it creates a right triangle with an angle of 32 with the horizontal line. Then displacement is
[tex]d_{x} = 147cos(32)[/tex] = 124.6631
[tex]d_{y}=147sin(32)[/tex] = 77.8981
Part (3):
The car goes down forming an angle below the horizontal. This trajectory also form a right triangle with an external angle. Knowing that alternate interior angles are congruent, the angle the right triangle forms with the horizontal line is 47°. So, displacement is
[tex]d_{x}=147cos(47)[/tex] = -100.2537
[tex]d_{y}=147sin(47)[/tex] = -107.509
Displacements in x and y are negative because they point towards the negative side of the reference.
To calculate total displacement:
[tex]d_{total}=\sqrt{d_{Tx}^{2}+d_{Ty}^{2}}[/tex]
Total displacement at each coordinate are:
[tex]d_{Tx}[/tex] = 202 + 124.6631 - 110.2537 = 226.41
[tex]d_{Ty}[/tex] = 77.8981 - 107.509 = -29.611
[tex]d_{total}=\sqrt{(226.41)^{2}+(-29.611)^{2}}[/tex]
[tex]d_{total}[/tex] = 228.3381
The car's displacement is 228.3381 ft.
For the direction, determine reference angle by:
[tex]\theta = tan^{-1}(\frac{d_{Ty}}{d_{Tx}} )[/tex]
[tex]\theta = tan^{-1}(\frac{29.611}{226.41} )[/tex]
θ = 7.46°
The direction of the car is at 7.46°.
