A ball is thrown up onto a roof, landing 4.50 s later at height h = 20.0 m above the release level. The ball's path just before landing is angled at θ = 54.0˚ with the roof. (a) Find the horizontal distance d it travels. What are the (b) magnitude and (c) angle (relative to the horizontal) of the ball's initial velocity?
To solve this problem, we need to consider the motion of the ball both horizontally and vertically.
(a) To find the horizontal distance the ball travels, we can use the formula:
d = v₀₁ * t
where v₀₁ is the horizontal component of the initial velocity and t is the time of flight.
Let's find v₀₁ first. Since the ball is thrown at an angle, we can find the horizontal component of the initial velocity using the formula:
v₀₁ = v₀ * cos(θ)
where v₀ is the initial velocity of the ball and θ is the angle with the roof.
Next, we can find the time of flight by using the formula for vertical motion:
h = v₀ * sin(θ) * t + (1/2) * g * t²
where h is the vertical displacement, g is the acceleration due to gravity (-9.8 m/s²), and t is the time of flight.
Rearranging the equation, we have:
t = (v₀ * sin(θ) ± √(v₀² * sin²(θ) + 2 * g * h)) / g
Since the ball is coming down after being thrown up, we can take the positive root.
Now, we have both v₀₁ and t. Plugging in these values into our first formula, we can calculate the horizontal distance d.
(b) To find the magnitude of the ball's initial velocity, we can use the Pythagorean theorem:
v₀ = √(v₀₁² + v₀₂²)
where v₀₂ is the vertical component of the initial velocity.
We can find v₀₂ using the formula:
v₀₂ = v₀ * sin(θ)
(c) To find the angle (relative to the horizontal) of the ball's initial velocity, we can use trigonometry:
θ₀ = tan⁻¹(v₀₂ / v₀₁)
Now, let's calculate the values.
First, plug in the values:
h = 20.0 m
θ = 54.0˚
g = 9.8 m/s²
Next, calculate the time of flight:
t = (v₀ * sin(θ) + √(v₀² * sin²(θ) + 2 * g * h)) / g
Then, find v₀₁:
v₀₁ = v₀ * cos(θ)
Now, calculate the horizontal distance:
d = v₀₁ * t
To find the magnitude of the initial velocity, use:
v₀ = √(v₀₁² + v₀₂²)
where v₀₂ = v₀ * sin(θ).
Finally, use trigonometry to find the angle θ₀:
θ₀ = tan⁻¹(v₀₂ / v₀₁)