A tennis ball, starting from rest at a height h = 2.10 m, rolls down the hill. At the end of the hill the ball becomes airborne, leaving at an angle of 27.0° with respect to the ground. Treat the ball as a thin-walled spherical shell (I = ⅔mr²), and determine the range x.
It's not 3.3972, or 9.78, or 1.7 and I can not figure out the answer.
To determine the range of the tennis ball, we need to consider its motion along the incline and its subsequent projectile motion after becoming airborne.
First, let's analyze the motion along the incline:
1. Calculate the initial speed:
The ball starts from rest at a height h = 2.10 m. We can find the initial speed using the conservation of mechanical energy.
Potential energy at the top of the hill = Kinetic energy at the bottom of the hill
mgh = (1/2)mv²
Since the mass m cancels out, we are left with:
gh = (1/2)v²
v² = 2gh
Substituting the given values:
v² = 2 * 9.8 m/s² * 2.10 m
v = √(2 * 9.8 * 2.10)
Now we have the initial speed v along the incline.
2. Calculate the time taken along the incline:
To find the time taken along the incline, we need to consider the acceleration of the ball.
Using the kinematic equation:
vf = vi + at
Since the ball starts from rest, the initial velocity vi is 0. The final velocity vf along the incline can be found using the equation:
vf² = vi² + 2ad,
Where a is the acceleration and d is the distance traveled along the incline.
Since the ball is rolling without slipping, the acceleration a can be related to the angular acceleration α:
a = rα.
Using the moment of inertia I = (2/3)mr² for a thin-walled spherical shell and the torque τ = Iα, we can find the angular acceleration α:
τ = rF = Iα
mgd = (2/3)mr²α,
g * d = (2/3)rα.
Now we can substitute the value of a = rα into the equation for vf:
vf² = 0 + 2 * (2/3)rg * d,
Since the final speed vf is equal to v, we can write:
v² = 4/3 * rg * d.
Substituting the values we know:
(√(2 * 9.8 * 2.10)) ² = 4/3 * (9.8 * d).
(2 * 9.8 * 2.10) = 4/3 * (9.8 * d),
2 * 2.10 * 3 = 4/3 * d.
Simplifying, we find:
d = (2 * 2.10 * 3) / (4/3).
d = (4.20 * 3) / (4/3).
d = 12.6 / (4/3).
d = 12.6 * (3/4).
d = 9.45 m.
3. Calculate the time of flight while in projectile motion:
After leaving the incline, the tennis ball will follow a projectile motion. The time of flight can be found using the formula:
t = 2 * (v * sinθ) / g,
where θ is the angle with respect to the ground.
Substituting the values:
t = 2 * (√(2 * 9.8 * 2.10) * sin(27°)) / 9.8.
Calculating this value, we find:
t ≈ 2.16 s.
4. Calculate the range x:
The range of the projectile motion can be determined using the formula:
x = v * cosθ * t.
Substituting the known values:
x = (√(2 * 9.8 * 2.10) * cos(27°)) * 2.16.
Calculating this value, we find:
x ≈ 4.87 m.
Therefore, the range of the tennis ball is approximately 4.87 meters.