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.
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?
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