Can I please get help on solving the structure of the next question?

Question

A golfer takes three strokes to putt a golf ball into a hole. On the first stroke, the ball moves 5.9 m due east. On the second, it moves 3.0 m at an angle 20° north of east. On the third, it moves 0.50 m due north. If the golfer had instead hit the ball directly into the hole on the first stroke, what would have been the magnitude (in m) and direction of the ball's displacement? Give the direction as a positive angle measured counterclockwise from due east.

Answer 1

So this is a vector addition problem. It is easiest to do these in terms of the math angle (counterclockwise from east) so

d1 = 5.9 m, 0°

d2 = 3m, 20°

d3 = .5m, 90°

The official approach is to add the components of the displacement vectors

d_T,x = d₁cosθ₁ + d₂cosθ₂ + d₃cos(θ₃) (you can plug and chug or you can be clever and take care of coordinate vectors yourself d_T,x = 5.9 + 3cos(20°)

d_T,y = same, but using sin of the angles or d_T,y = 3sin(20°) + .5

Now resolving these components back into (magnitude, angle) form:

d_T = sqrt(d_T,x² + d_T,y²) and θ_T = tan^-1(d_T,y/d_T,x) (note that you add 180º if d_T,x < 0)

Voila, Take care and please consider a tutor.

Answer 2

First let's get the directions in the problem into degrees. At the end of the problem it says to state the direction of the displacement as counterclockwise from due east. That means due east is 0°, which makes north 90°, west 180° and south 270°. In order to find the overall displacement for the golf ball, we need to know the sum of the "x" displacements and the sum of the "y" displacements of the ball.

For stroke 1: 5.9 m due east means 5.9 m at 0°

x displacement = (5.9 m)(cos 0°) = 5.9 m, y displacement = (5.9 m)(sin 0°) = 0 m

For stroke 2: 3.0 m at 20° north of east means 3.0 m at 20°

x displacement = (3.0 m)(cos 20°) = 2.82 m, y displacement = (3.0 m)(sin 20°) = 1.026 m

For stroke 3: 0.50 m due north means 0.5 m at 90°

x displacement = (0.5 m)(cos 90°) = 0 m, y displacement = (0.5 m)(sin 90°) = 0.5 m

The overall x displacement = 5.9 m + 2.82 m + 0 = 8.72 m

The overall y displacement = 0 m + 1.026 m + 0.5 m = 1.526 m

To find the magnitude of the magnitude of the overall displacement, use the Pythagorean Theorem:

d = ((8.72)² + (1.526)²))^1/2 = 8.85 m

To find the direction, take the inverse tangent of of the y and x displacements:

Θ = tan^-1(1.526/8.72) = 9.93°

Final answer for the overall displacement of the golf ball = 8.85 m @ 9.93°