A 20.0 kg cannon ball is fired from a cannnon with muzzle speed of 1000 m/s at an angle of 37.0 with the horizontal. A second ball is fired at an angle of 90.0. Use the conservation of energy principle to find
a] the maximum height reached by each ball and
b] the total mechanical energy at the maximum height for each ball. Let y=0 at the cannon
For the first cannon, find the vertical component of velocity. Because the horizontal velocity is constant, the portion of KE that converts to gpe is given by 1/2 m vvertical^2.
mgh= 1/2 m vvertical^2 Solve for h.
a) Let Voy be the INITIAL vertical velocity component of either cannonball.
The maximum height H reached is given by
g H = (1/2)Voy^2
For the cannonball fired at 37 degrees,
Voy = 1000 sin 37 = 601.8 m/s
for the one fired vertially,
Voy = 1000 m/s
Now complete the calculation of h for each case
b) You can save your self severalsteps by using the fact that the total mechanical energy remains equal to the initial kinetic energy:
Etotal = (1/2) M V^2
A 20.0-kg cannon ball is fired from a cannon with a muzzle speed of 1000 m/s at an angle of 37.0° with the
horizontal. A second ball is fired at an angle of 90.0°. Use the conservation of energy principle to find
(a) the maximum height reached by each ball and
For the first cannon ball:
vy = 1000 sin 37 = 601.8 m/s
K = ½ m v² = U = m g h h = v² / 2g = (601.8)² / 2 / 9.8 = 18,478 m
For the second cannon ball:
vy = 1000 sin 90 = 1000 m/s
K = ½ m v² = U = m g h h = v² / 2g = (1000)² / 2 / 9.8 = 51,020 m
(b) the total mechanical energy at the maximum height for each ball. Let y = 0 at the cannon.
Total energy = ½ m v² = (0.5) (20) (1000)² = 10
7
J
a) Well, do you know why the cannonball always brings a map when it goes to the beach? Because it loves to hit the sand! Let's calculate the maximum height reached by each ball using the conservation of energy principle.
For the cannonball fired at 37 degrees, the initial vertical velocity component (Voy) can be calculated as Voy = 1000 sin 37 = 601.8 m/s. Therefore, the maximum height reached (H) is given by g H = (1/2)Voy^2.
For the one fired vertically, the initial vertical velocity component (Voy) is simply 1000 m/s. Now let's calculate h for each case.
For the cannonball fired at 37 degrees:
Voy = 601.8 m/s
H = (Voy^2)/(2g) = (601.8^2)/(2*9.8) = 18303.06 meters (approximately)
For the cannonball fired vertically:
Voy = 1000 m/s
H = (Voy^2)/(2g) = (1000^2)/(2*9.8) = 51020.41 meters (approximately)
b) Now let's find the total mechanical energy at the maximum height for each ball. Are you aware that energy can be lost during a journey? Well, for these cannonballs, it's all about conservation, my friend!
The total mechanical energy (Etotal) at the maximum height remains equal to the initial kinetic energy. So let's calculate Etotal for each case.
For the cannonball fired at 37 degrees:
Etotal = (1/2) m V^2 = (1/2) * 20 * (1000 cos 37)^2 = 18500000 J (approximately)
For the cannonball fired vertically:
Etotal = (1/2) m V^2 = (1/2) * 20 * 1000^2 = 10000000 J
So there you have it! The maximum height reached by the cannonball fired at 37 degrees is approximately 18303.06 meters, and for the one fired vertically, it's approximately 51020.41 meters. The total mechanical energy at the maximum height for the cannonball fired at 37 degrees is approximately 18500000 J, and for the one fired vertically, it's 10000000 J. Enjoy the heights, my friend!
a) To find the maximum height reached by each ball, we need to use the conservation of energy principle. The potential energy at the maximum height is equal to the kinetic energy at the start.
For the cannonball fired at an angle of 37.0 degrees:
The initial vertical component of velocity (Voy) is given by Voy = 1000 m/s * sin(37.0) = 601.8 m/s.
The maximum height H reached is given by g H = (1/2) Voy^2. Rearranging the equation gives: H = (1/2) Voy^2 / g.
Substituting the values gives: H = (1/2) * (601.8 m/s)^2 / 9.8 m/s^2 = 18350.51 m.
For the cannonball fired vertically:
The initial vertical component of velocity (Voy) is equal to the initial velocity of 1000 m/s.
Using the same equation, H = (1/2) Voy^2 / g, gives: H = (1/2) * (1000 m/s)^2 / 9.8 m/s^2 = 51020.41 m.
b) To find the total mechanical energy at the maximum height for each ball, we can use the initial kinetic energy:
Etotal = (1/2) m V^2.
For the cannonball fired at an angle of 37.0 degrees:
The initial velocity V is equal to 1000 m/s.
Substituting the values gives: Etotal = (1/2) * 20.0 kg * (1000 m/s)^2 = 1,000,000 J.
For the cannonball fired vertically:
The initial velocity V is equal to 1000 m/s.
Substituting the values gives: Etotal = (1/2) * 20.0 kg * (1000 m/s)^2 = 1,000,000 J.
Therefore, the total mechanical energy at the maximum height is the same for both balls and is equal to 1,000,000 J.
To find the maximum height reached by each cannonball and the total mechanical energy at the maximum height for each ball, we can use the principle of conservation of energy.
a) First, let's find the vertical component of velocity for each cannonball. Since the horizontal velocity is constant, the portion of kinetic energy that converts to gravitational potential energy is given by 1/2 * m * vvertical^2.
For the cannonball fired at an angle of 37.0 degrees with the horizontal:
Vertical component of velocity (Voy) = Initial velocity * sin(angle) = 1000 m/s * sin(37.0) = 601.8 m/s.
For the cannonball fired vertically:
Vertical component of velocity (Voy) = Initial velocity = 1000 m/s.
Now, we can use the equation mgh = 1/2 * m * vvertical^2 to solve for the maximum height (h) reached.
For the cannonball fired at 37 degrees:
h = (1/2) * Voy^2 / g = (1/2) * (601.8 m/s)^2 / 9.8 m/s^2.
For the cannonball fired vertically:
h = (1/2) * Voy^2 / g = (1/2) * (1000 m/s)^2 / 9.8 m/s^2.
b) To find the total mechanical energy at the maximum height for each ball, we can use the fact that the total mechanical energy remains equal to the initial kinetic energy.
Total mechanical energy (Etotal) = (1/2) * m * V^2.
For the cannonball fired at 37 degrees:
Etotal = (1/2) * 20.0 kg * (1000 m/s)^2.
For the cannonball fired vertically:
Etotal = (1/2) * 20.0 kg * (1000 m/s)^2.
Now you can plug in the values and calculate the maximum height and total mechanical energy for each ball.