To determine the initial angle for a projectile to travel 100 feet, we use the range equation for projectile motion, convert feet to meters for compatibility with the acceleration due to gravity, and with the initial velocity value, we calculate the angle using algebraic manipulation.
To find the initial angle required to achieve exactly 100 feet in horizontal distance with a given initial velocity, we will use the projectile motion equation for range (R), which is R = (v0² sin(2θ))/g, where g is the acceleration due to gravity (approximately 9.8 m/s²), v0 is the initial velocity, and θ is the launch angle above the horizontal.
Given that our objective is to reach a horizontal distance (h) of 100 feet, we first need to convert feet to meters, since standard gravitational acceleration is in m/s. Thus, 100 feet is approximately 30.48 meters.
Now, substituting the known values into the equation, we get:
30.48 = (v0² sin(2θ))/9.8
Without the specific value of the initial velocity (v0), we cannot compute the exact angle. However, assuming the value of v0 is given, we would isolate sin(2θ) in the equation to solve for the angle, and then use the arcsine function (sin^-1) and algebraic manipulation to calculate the value of θ.
Lastly, it is important to note that if v0 was provided, there would typically be two possible angles that could result in the same range due to the symmetrical nature of projectile motion - one acute and one obtuse.
An example calculation would look like this if v0 was known:
- Convert 100 feet to meters to get 30.48 meters.
- Plug the value of v0 and 30.48 into the range equation, then solve for sin(2θ).
- Calculate θ by taking the inverse sine of the result, and divide by 2.