since the range is
R = (v^2 sin2θ)/g, I get
60^2 * sin(1.094)/16 = 200.0
Looks like you lost a factor of 2 there, since sin2θ = 2sinθcosθ
a. Assume the initial velocity is 60 ft/second. What initial angle will you need to ensure that the horizontal distance will be exactly 100 feet?
Here is my answer. Can anyone check if it's correct. Thank you.
60^2/16 sinᶿ cosᶿ=100
yields sin^2 ᶿ=.8889
theta=.547 radians
theta= 33.16 degrees
R = (v^2 sin2θ)/g, I get
60^2 * sin(1.094)/16 = 200.0
Looks like you lost a factor of 2 there, since sin2θ = 2sinθcosθ
h = (v0^2/16) * sinθ * cosθ
We know that h = 100 feet and v0 = 60 ft/second, so let's substitute these values into the equation:
100 = (60^2/16) * sinθ * cosθ
Now, we can simplify by evaluating 60^2/16:
100 = (3600/16) * sinθ * cosθ
100 = 225 * sinθ * cosθ
Let's divide both sides of the equation by 225:
100/225 = sinθ * cosθ
4/9 = sinθ * cosθ
Now, we can solve for sinθ:
sinθ = √(4/9)
Taking the square root, we get:
sinθ = 2/3
To find θ, we can use the inverse sine function:
θ = sin^(-1)(2/3)
Using a calculator, we find:
θ = 0.7297 radians
To convert this to degrees, we multiply by 180/π:
θ ≈ 41.81 degrees
Thus, the initial angle required to ensure a horizontal distance of exactly 100 feet is approximately 41.81 degrees.
h = (v0^2 / 16) * sin(theta) * cos(theta)
Replacing v0 with 60 ft/second and h with 100 feet, we get:
100 = (60^2 / 16) * sin(theta) * cos(theta)
Simplifying the equation, we have:
100 = 225 * sin(theta) * cos(theta)
Now, we need to solve for sin^2(theta) to eliminate both sin and cos terms in the equation. Divide both sides of the equation by 225:
sin^2(theta) * cos^2(theta) = 100 / 225
sin^2(theta) * cos^2(theta) = 0.4444
Now take the square root of both sides of the equation:
sin(theta) * cos(theta) = sqrt(0.4444)
To simplify further, remember the trigonometric identity sin(2theta) = 2sin(theta)cos(theta). We can rewrite the equation as:
sin(theta) * cos(theta) = sin(2theta) / 2
So now we have:
sin(2theta) / 2 = sqrt(0.4444)
Multiply both sides of the equation by 2 to isolate sin(2theta):
sin(2theta) = 2 * sqrt(0.4444)
Now, we can take the inverse sine of both sides:
2theta = arcsin(2 * sqrt(0.4444))
Divide both sides of the equation by 2:
theta = (1/2) * arcsin(2 * sqrt(0.4444))
Using a calculator, we find:
theta ≈ 33.16 degrees (rounded to two decimal places)
So, the initial angle that will result in a horizontal distance of exactly 100 feet is approximately 33.16 degrees. Your answer is correct!