Final Kinetic Energy and speed of three projectiles fired from a building's roof

[matthew_phys]

Homework Statement

Three different mass projectiles, mA > mB > mC, are launched from the top of a building at different angles with respect to the horizontal. Each particle has the same initial kinetic energy.

Which particle has the greatest kinetic energy just as it hits the ground? Why?
Which particle has the greatest speed just as it hits the ground? Why?

Homework Equations

(KE + PE)i = (KE + PE)f
KE = .5 mv^2
PE = mgh

The Attempt at a Solution

I don't know if it is the particle that has the highest mass m or attains the highest height h. Obviously, the one that attains the highest point will have the highest final velocity. vf = sqrt(2gh)

My teacher said that's wrong and now I don't know what to put.

 

[LawrenceC]

Hint: What can you say about the kinetic energy of each projectile as it reaches the same level as when it was launched? At that point, think about potential energy.

 

[matthew_phys]

Hmm that is very helpful. At the point where they were launched, the energy of each projectile would equal its initial kinetic energy. So the initial kinetic energy + mgh at that point would equal the total energy, so mA has the highest?

 

[LawrenceC]

Yes, you are correct.

Each projectile will have its speed increased by (2gh)^.5 when it hits the ground. That implies something about which one is going the fastest when it hits.

This is all the assistance I can give. I'll be away from a computer for the next few days.

 

[asteriatic]

I would like to clarify that the final kinetic energy and speed of a projectile will depend on a variety of factors, including the initial velocity, angle of launch, air resistance, and the height of the building. It is important to note that the initial kinetic energy of the projectiles is the same, but their final kinetic energy and speed will differ due to these factors.

Assuming that all other factors are constant, the projectile with the highest mass will have the greatest kinetic energy just as it hits the ground. This is because kinetic energy is directly proportional to mass, as seen in the equation KE = 0.5mv^2. Therefore, the particle with the higher mass (mA) will have a higher final kinetic energy compared to particles B and C.

However, when it comes to the greatest speed just as it hits the ground, it is not as straightforward. The particle that attains the highest point will have the highest final velocity, as stated in the equation vf = √(2gh). This means that particle C will have the highest speed just as it hits the ground. However, if we consider the angle of launch, the particle with the steepest angle (particle A) may have a higher speed due to the vertical component of its velocity. This illustrates that the final speed of a projectile is influenced by multiple factors and cannot be determined solely based on mass or height.

In conclusion, the particle with the highest mass (mA) will have the greatest kinetic energy just as it hits the ground, while the particle with the steepest angle (particle A) may have the greatest speed. However, the final speed of a projectile is a complex phenomenon and cannot be determined solely based on these factors.