A car is driven east for a distance of 43 km, then north for 20 km, and then in a direction 27° east of north for 25 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.
My answer for displacement is 72KM North-east. And 26 degree. so please help if possible, because this is my last try, and it been telling me i was wrong so far. Thanks
Well Amy, I solved it . Good luck to her
To determine the magnitude of the car's total displacement, we can use the Pythagorean theorem to find the resultant distance traveled.
(a) The car is driven 43 km east, 20 km north, and 25 km in a direction 27° east of north.
Using the east and west distance traveled, we can consider them as components of the displacement vector.
East component: 43 km
North component: 20 km + 25 km * sin(27°)
East component: 43 km
North component: 20 km + 25 km * 0.45399
≈ 20 + 11.35
≈ 31.35 km
Using the Pythagorean theorem:
Displacement (D) = sqrt(East component^2 + North component^2)
= sqrt(43^2 + 31.35^2)
≈ sqrt(1849 + 982.2225)
≈ sqrt(2831.2225)
≈ 53.17 km
Therefore, the magnitude of the car's total displacement from its starting point is approximately 53.17 km.
(b) To find the angle (from east) of the car's total displacement measured from its starting direction, we can use the tangent function.
Angle = arctan(North component / East component)
Angle = arctan(31.35 km / 43 km)
≈ arctan(0.728)
≈ 35.21°
So, the angle (from east) of the car's total displacement measured from its starting direction is approximately 35.21°.
To determine the car's total displacement, we can break down the car's motion into two components: the north-south displacement and the east-west displacement.
First, let's determine the east-west displacement. The car is driven east for a distance of 43 km, and then in a direction 27° east of north for 25 km. Using trigonometry, we can find the east-west displacement as follows:
east-west displacement = 43 km + 25 km * sin(27°)
Using the given values:
east-west displacement = 43 km + 25 km * sin(27°) ≈ 43 + 11.576 ≈ 54.576 km
Now let's determine the north-south displacement. The car is driven north for a distance of 20 km, and then in a direction 27° east of north for 25 km. Using trigonometry, we can find the north-south displacement as follows:
north-south displacement = 20 km + 25 km * cos(27°)
Using the given values:
north-south displacement = 20 km + 25 km * cos(27°) ≈ 20 + 21.555 ≈ 41.555 km
To determine the magnitude of the car's total displacement, we can use the Pythagorean theorem:
magnitude of displacement = sqrt((east-west displacement)^2 + (north-south displacement)^2)
Using the calculated values:
magnitude of displacement = sqrt((54.576 km)^2 + (41.555 km)^2) ≈ sqrt(2983.572 + 1724.950) ≈ sqrt(4708.522) ≈ 68.64 km
So, the answer to part (a) is that the magnitude of the car's total displacement is approximately 68.64 km.
To determine the angle of the car's total displacement measured from its starting direction, we can use trigonometry:
angle = atan(north-south displacement / east-west displacement)
Using the calculated values:
angle = atan(41.555 km / 54.576 km) ≈ atan(0.761) ≈ 37.08°
Therefore, the answer to part (b) is that the angle of the car's total displacement measured from its starting direction is approximately 37.08° from east.
It's important to note that the direction should be specified counterclockwise from the east direction. So, the final answer would be approximately 72.92° northeast (37.08° + 45°).