A golfer hits a shot to a green that is elevated 2.60 m above the point where the ball is struck. The ball leaves the club at a speed of 16.5 m/s at an angle of 30.0˚ 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 solve this problem, we need to break it down into different stages: the initial launch of the ball, its ascent, and its descent.
Let's start with the initial launch:
Given:
Initial speed, v₀ = 16.5 m/s
Launch angle, θ = 30.0°
We can decompose the initial velocity into horizontal and vertical components:
Horizontal component: v₀ * cos(θ)
Vertical component: v₀ * sin(θ)
Now let's find the time it takes for the ball to reach its maximum height:
Using the equation for vertical motion:
Δy = v₀y * t + 0.5 * a * t²
Where:
Δy = vertical displacement
v₀y = vertical component of initial velocity
a = acceleration due to gravity (approximately -9.8 m/s²)
t = time
Since the ball reaches its maximum height, its vertical displacement is zero. Therefore, the equation becomes:
0 = (v₀ * sin(θ)) * t + 0.5 * (-9.8) * t²
Simplifying this equation, we get:
4.9 * t² = v₀ * sin(θ) * t
Moving the 't' terms to one side gives us:
4.9 * t² - v₀ * sin(θ) * t = 0
Now we can use the quadratic formula to solve for 't':
t = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 4.9, b = -v₀ * sin(θ), and c = 0.
t = (-(-v₀ * sin(θ)) ± √((-v₀ * sin(θ))² - (4 * 4.9 * 0))) / (2 * 4.9)
Now we can calculate 't'.
Once we have 't', we can calculate the maximum height reached by the ball:
Δy = (v₀ * sin(θ)) * t + 0.5 * (-9.8) * t²
Now let's find the total time of flight:
We know that the time it takes to reach the maximum height is 't', and since the ball reaches its maximum height and falls back down, the total time of flight, t_total, is two times 't'.
t_total = 2 * t
Finally, we can find the speed of the ball just before it lands.
To do this, we need to calculate the horizontal and vertical components of the velocity at the time of impact.
Horizontal component: v_horizontal = v₀ * cos(θ) (remains constant throughout the motion)
Vertical component: v_vertical = v₀ * sin(θ) - g * t
The magnitude of the final velocity can then be found using the Pythagorean theorem:
v_final = √(v_horizontal² + v_vertical²)