Recall that the equation for the horizontal distance h in feet of a projectile with initial velocity v0 and initial angle theta is given by h=v0^2/16 sin theta cos theta .

Question

Recall that the equation for the horizontal distance h in feet of a projectile with initial velocity v0 and initial angle theta is given by h=v0^2/16 sin theta cos theta .

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

Answer 1

furry sack of balls

Answer 2

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

Answer 3

To solve for the initial angle θ, we will rearrange the equation:

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.

Answer 4

To solve for the initial angle theta that will result in a horizontal distance of 100 feet, we can start by substituting the given values into the equation:

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!