A golfer hits a shot to a green that is elevated 3 m above the point where the ball is struck. The ball leaves the club at a speed of 15.6 m/s at an angle of 41.4° above the horizontal. It rises to its maximum height and then falls down to the green. Ignoring air resistance, find the speed of the ball just before it lands.

To find the speed of the ball just before it lands, we can break down the motion of the ball into horizontal and vertical components.

First, let's calculate the time it takes for the ball to reach its maximum height. The initial vertical velocity is given by V₀ * sin(θ), where V₀ is the initial speed of the ball (15.6 m/s) and θ is the angle of the shot (41.4°). The acceleration due to gravity is -9.8 m/s². The time taken to reach maximum height can be found using the equation:

V = V₀ + at

where V is the final vertical velocity at maximum height (0 m/s). Rearranging the equation, we get:

0 = V₀ * sin(θ) - 9.8 * t

Solving for t, we have:

t = V₀ * sin(θ) / 9.8

Next, we can calculate the time it takes for the ball to reach the ground from its maximum height. The total vertical distance traveled by the ball is 3 m (elevation of the green). Using the equation of motion:

s = V₀ * t + (1/2) * a * t²

where s is the vertical distance, V₀ is the initial vertical velocity, t is the time, and a is the acceleration due to gravity, we can solve for t. Plugging in the values, we have:

3 m = V₀ * sin(θ) * t + (1/2) * (-9.8 m/s²) * t²

Rearranging and simplifying, we get a quadratic equation:

-4.9t² + V₀ * sin(θ) * t + 3 = 0

Using the quadratic formula, we can solve for t. The positive root will give us the time it takes for the ball to reach the ground from its maximum height.

Once we have the time, we can calculate the horizontal distance traveled by the ball. The horizontal velocity remains constant throughout the motion and is given by V₀ * cos(θ), where V₀ is the initial speed of the ball and θ is the angle of the shot.

Finally, we can determine the speed of the ball just before it lands using the horizontal distance traveled and the time taken to reach the ground.

Performing these calculations will give us the answer to the question.