A58.0-kg projectile is fired at an angle of 30.0° above the horizontal with an initial speed of 140 m/s from the top of a cliff 132 m above level ground, where the ground is taken to be y = 0. (a) what is the initial total mechanical energy of the projectile? (give your answer to at least three significant figures.) 4060 j (b) suppose the projectile is traveling 99.2 m/s at its maximum height of y = 336 m. how much work has been done on the projectile by air friction? j (c) what is the speed of the projectile immediately before it hits the ground if air friction does one and a half times as much work on the projectile when it is going down as it did when it was going up? m/

(a)

The total mechanical energy of the projectile at the beginning is the sum of the initial kinetic energy (K) and potential energy (U):

The initial kinetic energy is:

where m = 58.0 kg is the mass of the projectile and is the initial speed. Substituting,

The initial potential energy is given by

where g=9.8 m/s^2 is the gravitational acceleration and h=132 m is the height of the cliff. Substituting,

So, the initial mechanical energy is

(b)

We need to calculate the total mechanical energy of the projectile when it reaches its maximum height of y=336 m, where it is travelling at a speed of v=99.2 m/s.

The kinetic energy is

while the potential energy is

So, the mechanical energy is

And the work done by friction is equal to the difference between the initial mechanical energy of the projectile, and the new mechanical energy:

And the work is negative because air friction is opposite to the direction of motion of the projectile.

(c) 88.1 m/s

The work done by air friction when the projectile goes down is one and a half times (which means 1.5 times) the work done when it is going up, so:

When the projectile hits the ground, its potential energy is zero, because the heigth is zero: h=0, U=0. So, the projectile has only kinetic energy:

E = K

The final mechanical energy of the projectile will be the mechanical energy at the point of maximum height plus the work done by friction:

And this is only kinetic energy:

So, we can solve to find the final speed: