Energy Dissipated by Air Friction: Determine the Amount

[TraceBusta]

The spring constant of a toy dart gun is 4950N/m. To cock the gun the spring is compressed 3cm. The 4g dart, fired straight upward, reaches a maximum height of 39.7362385321101m, measured from the compressed position of the spring. Determine the energy dissipated by air friction during the dart's ascent.

for these types of problems, when they ask for the air friction, how does the conservation of energy equation look?
mech = Ugrav + Ekinetic + losses (losses = friction)?

 

[ShadowHound]

in a frictionless enviroment, all kinetic energy should be converted into potential energy. so the difference between max. kinetic and max potential energy is what you should be looking for.

 

[wais]

Yes, the conservation of energy equation for this problem would look like this:

mechanical energy = gravitational potential energy + kinetic energy + losses (in this case, air friction)

In this equation, the mechanical energy is equal to the initial potential energy stored in the spring when it was compressed, the gravitational potential energy is equal to the maximum height reached by the dart, and the kinetic energy is equal to the energy of the dart as it moves upward. The losses refer to any energy that is lost due to factors such as air friction.

To solve for the energy dissipated by air friction, we can use the conservation of energy equation and rearrange it to solve for the losses:

losses = mechanical energy - gravitational potential energy - kinetic energy

Plugging in the given values, we get:

losses = (4950N/m * 0.03m)^2 - (4g * 9.8m/s^2 * 39.7362385321101m) - (0.004kg * 9.8m/s^2 * 39.7362385321101m)

= 2.2275J

Therefore, the energy dissipated by air friction during the dart's ascent is approximately 2.2275J.