During the rodeo, a clown runs 11m north, turns 24º east of north, and runs 2.1m. Then, after waiting for the bull to come near, the clown turns due east and runs 8.3, to exit the arena.
What is the magnitude of the clown's total displacement?
How many degrees east of north is the clown's total displacement?
D=11N+2.1*cos24 N +2.1sin24E + 8.3E
add like terms.
displacement= sqrt (xxx^2 N + yyyy^2E) you figure xxxx and yyyy from above.
angle=arctan yyyy/xxxxx E of N
All angles are measured CW from +y-axis.
Disp. = 11m[0o] + 2.1m[24o] + 8.3m[90o].
X = 11*sin 0 + 2.1*sin24 + 8.3*sin90 = 9.15 m.
Y = 11*Cos 0 + 2.1*Cos24 + 8.3*Cos90 = 12.92 m.
Disp. = X + Yi = 9.15 + 12.92i = 15.83m[35.3o] N. of E.
Tan A = X/Y.
A = 35.3o
Correction: A = 35.3o E. of N.
To find the magnitude of the clown's total displacement, we can use the Pythagorean theorem. The clown's movement can be broken down into two components: 1) the north-south component, and 2) the east-west component.
1) North-south component:
The clown runs 11m north and then 2.1m. The north-south component is the sum of these two distances, which is 11m + 2.1m = 13.1m.
2) East-west component:
The clown turns 24 degrees east of north and runs 8.3m due east. We can find the east-west component by multiplying the distance by the cosine of the angle. So, the east-west component is 8.3m * cos(24º) = 7.44m.
Now, we can find the magnitude of the clown's displacement using the Pythagorean theorem:
Magnitude of displacement = sqrt((north-south component)^2 + (east-west component)^2)
Magnitude of displacement = sqrt((13.1m)^2 + (7.44m)^2)
Magnitude of displacement = sqrt(171.61m^2 + 55.3536m^2)
Magnitude of displacement = sqrt(226.9636m^2)
Magnitude of displacement ≈ 15.05m
Therefore, the magnitude of the clown's total displacement is approximately 15.05 meters.
To determine the angle east of north, we can use trigonometry. The angle can be found using the arctangent function:
Angle = arctan(east-west component / north-south component)
Angle = arctan(7.44m / 13.1m)
Angle ≈ 0.52 radians
To convert this angle to degrees, we multiply it by 180/π:
Angle (in degrees) ≈ (0.52 radians) * (180/π) ≈ 29.78º
Therefore, the clown's total displacement is approximately 15.05 meters with an angle of approximately 29.78º east of north.