A placekicker is about to kick a field goal. The ball is 20.1 m from the goalpost. The ball is kicked with an initial velocity of 18.7 m/s at an angle θ above the ground. Between what two angles, θ1 and θ2, will the ball clear the 3.12-m-high crossbar? Give your answers as (a) the smaller angle and (b) the larger angle. (Hint: The following trigonometric identities may be useful:
sec θ = 1/(cos θ) and sec2θ = 1 + tan2θ.)
To determine between what two angles the ball will clear the crossbar, we can break down the motion into vertical and horizontal components.
First, let's find the time it takes for the ball to reach the crossbar. To do this, we'll use the vertical component of motion.
The vertical distance traveled by the ball can be calculated using the equation:
y = v₀y * t + (1/2) * a * t²
Where:
y = vertical distance (3.12 m)
v₀y = initial vertical velocity (v₀ * sin(θ))
a = acceleration due to gravity (-9.8 m/s²)
t = time
Using the equation, we can rearrange it to solve for time:
t = (v₀y ± √(v₀y² - 2 * a * y)) / a
Since we want to find the time it takes for the ball to reach a height of 3.12 m, we'll use the positive square root form of the equation.
Now, let's calculate the vertical velocity component:
v₀y = v₀ * sin(θ)
Given that v₀ = 18.7 m/s, we can calculate v₀y using the value of θ.
Next, we'll calculate the time it takes for the ball to reach the crossbar using the equation mentioned above.
Once we have the time, we can calculate the horizontal distance traveled by the ball using the equation:
x = v₀x * t
Where:
x = horizontal distance
v₀x = initial horizontal velocity (v₀ * cos(θ))
t = time
The horizontal velocity component, v₀x, can be found using the value of θ.
Now we have the horizontal and vertical distances traveled by the ball. To clear the crossbar, the vertical distance should be greater than or equal to the height of the crossbar. However, we need to check if the horizontal distance is within the range of 20.1 m.
Using these calculations, you can substitute different values of θ within a suitable range (e.g., 0° to 90°) and check if the conditions are met. This way, you can find the angles θ1 and θ2 between which the ball will clear the crossbar.