Answer Mech Energy of 32kg Cannon Ball at Max Height

[therest]

Homework Statement

A 32 kg cannon ball is fired from a cannon with muzzle speed of 1360 m/s at an angle of 44◦ with the horizontal. The acceleration of gravity is 9.8 m/s² . What is the total mechanical energy at the maximum height of the ball? Answer in units of J.

Homework Equations

Mechanical energy = KE + U_G + U_S (ignore spring potential energy)
U_G = mgh
KE = (1/2)mv²

The Attempt at a Solution

I actually thought I understood the concepts behind this one fine, but I keep getting the wrong answer.

First I found the height:
Theoretically, mechanical energy is conserved, so E at launch = E at top of flight (w/ max height)
KE + U_G = KE + U_G
no U_G at launch, and no KE at top of flight in the y-direction -- motion is changing direction -- so
KE in the vertical direction = U_G
(1/2)(32 kg)(1360sin44 m/s) = (32 kg)(9.8 m/s²)h
h = 45536.98701 m

Then, I attempted to find the mechanical energy at the top of the flight.
Known facts:
* the mass has a vertical acceleration and a horizontal velocity of 1360cos44 m/s
* the mass has both kinetic energy (it is moving) and potential gravitational energy.
* I guessed that the total mechanical energy would be the resultant of these two using the Pythagorean theorem since the two energies are perpendicular to each other.

E at top = KE in the x-direction + U_G in the y-direction
KE = (1/2)(32)(1360cos44)² = 15313200.87 J
U_G = (32)(9.8)(45536.98701) = 14280399.13 J

KE² + U_G² = resultant²
resultant = 20938574.93 J

But... that's the wrong answer.
Do the directions of the vectors not matter? Can someone provide an explanation for this?

 

[ehild]

Energy is scalar, has no direction, you simply have to add the different kinds of energy.

"KE² + UG² = resultant²" this is wrong.

As for your method, you overcomplicate the calculations. Conservation of the mechanical energy means that the total energy E= KE + PE at the muzzle is the same as at the maximum height.

ehild

 

[kerimek]

Your approach to finding the mechanical energy at the maximum height of the cannon ball is correct, but there are a few mistakes in your calculations. First, the vertical component of the velocity should be 1360sin44 m/s, not 1360cos44 m/s. This is because the angle of 44 degrees is measured from the horizontal, not the vertical. Then, when finding the kinetic energy in the x-direction, you should use the horizontal velocity, which is 1360cos44 m/s, not the vertical velocity. Finally, when finding the total mechanical energy, you should add the kinetic energy in the x and y directions, not take the Pythagorean theorem of these two energies. So the correct calculation would be:

KE in the x-direction = (1/2)(32)(1360cos44)^2 = 15313200.87 J
KE in the y-direction = (1/2)(32)(1360sin44)^2 = 12645421.42 J
UG = (32)(9.8)(45536.98701) = 14280399.13 J

Total mechanical energy = KE in the x-direction + KE in the y-direction + UG = 15313200.87 J + 12645421.42 J + 14280399.13 J = 42239021.42 J

So the total mechanical energy at the maximum height of the cannon ball is 42239021.42 J.