A student fires a 70kg cannonball as shown. The cannon exerts an average force of 67000N while inside the .76m long barrel. What’s the max height?
To calculate the maximum height, we need to use the principle of conservation of mechanical energy.
The initial energy of the cannonball is purely kinetic energy given by the equation:
KE = (1/2)mv^2
where m is the mass of the cannonball and v is its velocity. The velocity can be obtained using Newton's second law of motion:
F = ma
where F is the force exerted by the cannon and a is the acceleration. Rearranging the equation, we have:
a = F/m
Substituting the given values: m = 70 kg and F = 67,000 N:
a = 67,000 N / 70 kg = 957.14 m/s^2
To find the final velocity (v), we can use the equation:
v^2 = u^2 + 2as
where u is the initial velocity (which is zero as the cannonball starts from rest) and s is the distance traveled in the barrel (0.76 m).
v^2 = 0 + 2(957.14 m/s^2)(0.76 m)
v^2 = 1458.1373 m^2/s^2
Taking the square root of both sides, we find:
v ≈ 38.18 m/s
Now, we can calculate the maximum height (H) using the conservation of mechanical energy. At the maximum height, all the kinetic energy of the cannonball is converted into gravitational potential energy:
KE = PE
(1/2)mv^2 = mgh
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the maximum height. Rearranging the equation gives:
h = (1/2)v^2 / g
Substituting the values: v ≈ 38.18 m/s and g ≈ 9.8 m/s^2:
h = (1/2)(38.18 m/s)^2 / 9.8 m/s^2
h ≈ 74.14 m
Therefore, the maximum height reached by the cannonball is approximately 74.14 meters.