When a projectile leaves a starting point at an angle of elevation of theta with a velocity v, the horizontal distance it travels is determined by:
d=v^2/(32) sin2theta
Where d is measured in feet and v in feet per second.
An outfielder throws the ball at a speed of 75 miles per hour to the catcher who is 200 feet away. At what angle of elevation was the ball thrown?
sub in the values
200 = 75^2/2 sin^2 Ø
400/5625 = sin^2 Ø
sin Ø = √(400/5625) = 20/75 = 1/3
Ø = 19.5°
The answer says:16.0 or 74.0 degrees :|
To find the angle of elevation at which the ball was thrown, we can rearrange the formula:
d = (v^2)/(32) * sin(2θ)
Considering the values given:
d = 200 feet
v = 75 miles per hour
First, we need to convert the velocity from miles per hour to feet per second:
1 mile = 5280 feet
1 hour = 3600 seconds
So, 75 miles per hour = 75 * 5280 feet / 3600 seconds ≈ 110 feet/second.
Now, we can substitute the values into the formula and solve for θ:
200 = (110^2)/(32) * sin(2θ)
Simplifying,
200 = 12100/32 * sin(2θ)
Multiplying both sides by 32:
6400 = 12100 * sin(2θ)
Now, we can isolate sin(2θ) by dividing both sides by 12100:
sin(2θ) = 6400 / 12100
Taking the inverse sine:
2θ = arcsin(6400/12100)
Calculating the inverse sine:
2θ ≈ 30.6 degrees
Finally, dividing both sides by 2:
θ ≈ 15.3 degrees (rounded to one decimal place)
Therefore, the ball was thrown at an angle of elevation of approximately 15.3 degrees.
To solve this problem, we need to rearrange the given formula to solve for theta, the angle of elevation.
The formula is:
d = v² / (32 * sin(2theta))
Given information:
- d = 200 feet (the horizontal distance traveled by the projectile)
- v = 75 miles per hour
First, we need to convert the velocity, given in miles per hour, to feet per second.
1 mile = 5280 feet
1 hour = 3600 seconds
So, 75 miles per hour = (75 * 5280) feet / 3600 seconds = 110 feet per second.
Now we can substitute the values into the formula:
200 = (110)² / (32 * sin(2theta))
Let's simplify further:
200 = 12100 / (32 * sin(2theta))
Multiply both sides of the equation by (32 * sin(2theta)) to isolate the sin(2theta):
(32 * sin(2theta)) * 200 = 12100
6400 * sin(2theta) = 12100
Divide both sides of the equation by 6400:
sin(2theta) = 12100 / 6400
sin(2theta) ≈ 1.8906
To solve for 2theta, we need to use the arcsin function:
2theta = arcsin(1.8906)
Now, we divide by 2 to solve for theta:
theta = arcsin(1.8906) / 2
Using a calculator, we find:
theta ≈ 39.9 degrees
Therefore, the ball was thrown at an angle of elevation of approximately 39.9 degrees.