A placekicker is about to kick a field goal. The ball is 26.6 m from the goal post. The ball is kicked with an initial velocity of 19.0 m/s at an angle θ above the ground. Between what two angles, θ1 and θ2, will the ball clear the 2.8-m-high crossbar? (Hint: The following trigonometric identities may be useful sec θ = 1/(cos θ) and sec2 θ = 1 + tan2 θ.)

thbrtyhn

To determine the range of angles θ between which the ball will clear the 2.8-m-high crossbar, we need to solve the problem in two parts:

1. Determine the maximum height the ball reaches.
2. Determine the angle θ at which the ball reaches the crossbar height.

Let's start with the first part:

1. Determine the maximum height the ball reaches:

The vertical motion of the ball can be described using the equation:
h = v₀y * t - (1/2) * g * t²
where:
- h is the height
- v₀y is the vertical component of the initial velocity (v₀) which is v₀y = v₀ * sin(θ)
- t is the time of flight
- g is the acceleration due to gravity (approximately 9.8 m/s²)

At the maximum height, the vertical velocity component (v_y) is zero. We can use this information to find the time of flight (t):

v_y = v₀y - g * t_max = 0

From the above equation, we can find t_max as:
t_max = v₀y / g

Now we can substitute t_max into the equation for the maximum height (h) to find the value.

2. Determine the angle θ at which the ball reaches the crossbar height:

The horizontal distance (x) covered by the ball can be determined using the equation:
x = v₀x * t
where:
- v₀x is the horizontal component of the initial velocity (v₀) which is v₀x = v₀ * cos(θ)

To find the final angle, we substitute the known values into the equation for the maximum height (h) from step 1:
h = v₀y * t_max - (1/2) * g * t_max²

Since the crossbar height is 2.8 m, we can set up the equation:
h = 2.8 m

Now, we solve this equation by substituting values from step 1 and solving for θ.

By solving the two parts of the problem, we can find the range of angles (θ1 and θ2) between which the ball will clear the 2.8-m-high crossbar.