A football kicker can give the ball an initial speed of 31 m/s. What are the (a) least and (b)

greatest elevation angles at which he can kick the ball to score a field goal from a point 49 m in
front of goalposts whose horizontal bar is 3.44 m above the ground?

To determine the least and greatest elevation angles at which the kicker can kick the ball to score a field goal, we can use basic kinematic equations of motion and principles of projectile motion.

Let's break down the problem:

Given:
Initial speed of the ball (v0) = 31 m/s
Distance from the point of kick to the goalposts (horizontal range, R) = 49 m
Height of the goalposts' horizontal bar (h) = 3.44 m

To calculate the least and greatest elevation angles, we need to find the corresponding launch angles (θ) at which the ball would travel the required horizontal distance and clear the goalposts.

Step 1: Resolve Initial Velocity
To solve this problem, we need to resolve the initial velocity into horizontal and vertical components.

The horizontal component (v0x) remains constant throughout the motion and is given by:
v0x = v0 * cosθ

The vertical component (v0y) changes due to the effect of gravity and is given by:
v0y = v0 * sinθ

Step 2: Calculate Time of Flight
Once we have the vertical component, we can calculate the time of flight (t) of the projectile. The time it takes for the ball to reach the goalposts is the same as the time it takes for the ball to fall back down to the ground.

Using the equation:
h = (1/2) * g * t^2

where g is the acceleration due to gravity (approximately 9.8 m/s^2), we can solve for t.

Step 3: Calculate Maximum Height Reached
Next, we can calculate the maximum height (H) reached by the projectile. The maximum height is the halfway point between the initial and final vertical position of the ball.

Using the relation:
H = (v0y)^2 / (2 * g)

Step 4: Calculate Launch Angles
Now that we have the time of flight (t) and the maximum height (H), we can calculate the launch angles (θ).

The least and greatest launch angles can be obtained using the following formulas:

(a) Least Elevation Angle (θ_1):
θ_1 = arctan((H - h) / R)

(b) Greatest Elevation Angle (θ_2):
θ_2 = arctan((H + h) / R)

Step 5: Calculate the Actual Angles
Finally, we can calculate the actual least and greatest elevation angles by substituting the values into the equations above.

Please note that in trigonometry, inverse tangent functions (arctan) return an angle in radians, which can be converted to degrees if needed.

By following these steps, you should be able to find the least and greatest elevation angles at which the kicker can score a field goal.