A person was shot vertically upward from a cannon with a mass 52 kg and a speed of 35 m/s. a) what was the height according to conservation of energy and b) what was the person's total energy? (TME never changes!)

a) used TMEo=TMEf
height= 63 m.

b.)Would I use PE=mgh

I get 62.5 using g=9.8

Agreed.

a) h = 62.5 m
= 6.3 x 10^1 m

b) PE = mgh
=(52)(9.8)(62.5)
= 31850 j
= 3.2 x 10^4 J

To calculate the height reached by the person using conservation of energy, you can use the equation:

TMEo = TMEf

TME represents the total mechanical energy, which is the sum of kinetic energy (KE) and potential energy (PE).

The initial TME (TMEo) consists only of kinetic energy:

TMEo = 1/2 * mass * velocity^2

Plugging in the values given:

TMEo = 1/2 * 52 kg * (35 m/s)^2

TMEo = 1/2 * 52 kg * 1225 m^2/s^2

TMEo = 31850 J

The final TME (TMEf) consists of potential energy at maximum height, as the person's speed becomes zero:

TMEf = PE

To calculate potential energy, you can use the equation:

PE = mass * gravity * height

Where mass is the mass of the person, gravity is the acceleration due to gravity (approximately 9.8 m/s^2), and height is the unknown value we are trying to find.

Plugging in the known values:

TMEf = 52 kg * 9.8 m/s^2 * height

TMEf = 509.6 N * height

Since TME never changes, we can write:

TMEo = TMEf

31850 J = 509.6 N * height

Solving for height:

height = 31850 J / 509.6 N

height ≈ 62.527 m

So, according to the conservation of energy, the height reached by the person is approximately 62.527 meters.

Moving on to part b, to calculate the person's total energy (TME), we can use the equation:

TME = PE + KE

Where TME is the total mechanical energy, PE is the potential energy, and KE is the kinetic energy.

Since we have already calculated the potential energy (PE) using the formula PE = mgh, we can add it to the initial kinetic energy (KE) to find the total energy (TME).

Plugging in the values:

PE = mgh = 52 kg * 9.8 m/s^2 * 62.527 m

PE ≈ 31,423 J

KE = 1/2 * mass * velocity^2

KE = 1/2 * 52 kg * (35 m/s)^2

KE = 1/2 * 52 kg * 1225 m^2/s^2

KE ≈ 31,850 J

TME = PE + KE

TME ≈ 31,423 J + 31,850 J

TME ≈ 63,273 J

Therefore, the person's total energy (TME) is approximately 63,273 Joules.