Answer:
a) The maximum height of the first ball is 1.85 × 10⁴ m.
The maximum height reached by the second ball is 5.10 × 10⁴ m.
b) The mechanical energy of both balls is 1.00 × 10⁷ J at every moment.
Explanation:
Hi there!
At the cannon, the gravitational potential energy (PE) is zero because the height is zero:
PE = m · g · h
Where:
m = mass of the ball.
g = acceleration due to gravity (9.81 m/s²).
h = height.
The kinetic energy (KE) when the first ball is fired can be calculated as follows:
KE = 1/2 · m · v²
Where:
m = mass of the ball
v = velocity of the ball.
Then, the initial kinetic energy will be:
KE = 1/2 · 20.0 kg · (1000 m/s)²
KE = 1.00 × 10⁷ J
The mechanical energy (E) is constant because of the conservation of energy and can be expressed as the sum of the potential plus the kinetic energy:
E = PE + KE
E = 0 J + 1.00 × 10⁷ J = 1.00 × 10⁷ J
At the maximum height, the velocity of the ball will be given by the following vector:
v = (v0 · cos θ, 0)
(see the attached figure to understand why the y-component of the velocity is zero at its maximum height. Also notice that since the only force that acts on the ball is the force of gravity, there is no horizontal acceleration, thus, the x-component of the velocity is constant).
Where:
v0 = initial velocity of the ball.
θ = launching angle.
The magnitude of the velocity at the maximum height is v0 · cos θ.
Then, the kinetic energy at the maximum height can be calculated as follows:
KE = 1/2 · 20.0 kg · (1000 m/s · cos 37°)²
KE = 6.38 × 10⁶ J
Since the mechanical energy is constant, we can calculate the gravitational potential energy:
E = PE + KE
E - KE = PE
1.00 × 10⁷ J - 6.38 × 10⁶ J = PE
PE = 3.62 × 10⁶ J
Then we can solve the equation of PE for the height:
PE = m · g · h
3.62 × 10⁶ J = 20.0 kg · 9.81 m/s² · h
h = 3.62 × 10⁶ J / (20.0 kg · 9.81 m/s²)
h = 1.85 × 10⁴ m
The maximum height of the first ball is 1.85 × 10⁴ m
For the second ball, the kinetic energy at the maximum height will be zero because it is a vertical shot, there is no x-component of the velocity.
Then, at the maximum height the potential energy will be equal to the mechanical energy:
PE = E - KE
PE = 1.00 × 10⁷ J - 0 J
Using the equation of potential energy:
PE = m · g · h
1.00 × 10⁷ J = 20.0 kg · 9.81 m/s² · h
1.00 × 10⁷ J / (20.0 kg · 9.81 m/s²) = h
h = 5.10 × 10⁴ m
The maximum height reached by the second ball is 5.10 × 10⁴ m
b) The mechanical energy for each ball will be the same at every moment because of the law of conservation of energy. Initially, both balls have the same kinetic energy that is being continuously transformed into gravitational potential energy until the balls reach the maximum height. The mechanical energy of both balls is 1.00 × 10⁷ J.
