A boat travels at 15 m/s in a direction 45° east of north. The boat then turns and travels at 18 m/s in a direction 5° north of east.
What is the magnitude of the boat’s resultant velocity? Round your answer to the nearest whole number.
What is the direction of the boat’s resultant velocity? Round your answer to the nearest whole degree.
To find the magnitude of the boat's resultant velocity, we can use the law of cosines. The magnitude of the resultant velocity can be calculated using the formula:
Resultant velocity = sqrt((velocity1)^2 + (velocity2)^2 + 2 * velocity1 * velocity2 * cos(angle1 - angle2))
In this case, velocity1 is 15 m/s, velocity2 is 18 m/s, angle1 is 45°, and angle2 is 5°.
Plugging in the values, we have:
Resultant velocity = sqrt((15)^2 + (18)^2 + (2 * 15 * 18 * cos(45° - 5°)))
Calculating the value inside the square root:
Resultant velocity = sqrt(225 + 324 + (2 * 15 * 18 * cos(40°)))
Now, calculate cos(40°):
cos(40°) = 0.766
Now, plug in the value of cos(40°) and calculate:
Resultant velocity = sqrt(225 + 324 + (2 * 15 * 18 * 0.766))
Resultant velocity = sqrt(225 + 324 + 412.2)
Resultant velocity = sqrt(961.2)
Resultant velocity ≈ 30.98 m/s
Therefore, the magnitude of the boat's resultant velocity is approximately 31 m/s.
To find the direction of the boat's resultant velocity, we can use the tangent function. The direction can be calculated using the formula:
Direction = arctan(velocity2 * sin(angle2) / (velocity1 + velocity2 * cos(angle2)))
In this case, velocity1 is 15 m/s, velocity2 is 18 m/s, angle1 is 45°, and angle2 is 5°.
Plugging in the values, we have:
Direction = arctan(18 * sin(5°) / (15 + 18 * cos(5°)))
Calculating the value inside the arctan function:
Direction = arctan(18 * 0.087 / (15 + 18 * 0.996))
Direction = arctan(1.566 / 32.928)
Now, calculate arctan(1.566 / 32.928):
arctan(1.566 / 32.928) ≈ 2.832°
Therefore, the direction of the boat's resultant velocity is approximately 3°.
Hence, the magnitude of the boat's resultant velocity is 31 m/s, and the direction is 3°.
Using the usual trig angles (not compass headings),
15 cis45° + 18cis5° = 28.54+12.18i
= 31.03 cis23.11°