Can Work-Energy Be Used to Determine Maximum Projectile Height?

[whiskeySierra]

Please forgive me if the answer to this question is obvious, I studied Political Science for three years in undergrad before I realized 'real' science was far more rewarding.

My question (not homework):

I understand that Work-Energy can be used for projectile motion, if what you're after is not time dependent. I also find that working with W.E. is more intuitive for me than the kinematics equations.

Say I have a projectile that is launched at some angle < 90°, at some velocity and I want to know what the maximum height the projectile reaches in its flight. We will assume no air resistance and that height initial = height final.

So figuring out the maximum height of the projectile with kinematics is easy enough, but with W.E. it's not so straight forward.

I know that maximum height along the path would occur where PE has the highest value along the entire path. KE would never be zero because the projectile never stops moving, only slows a little at maximum height.

So how (if possible) can I use W.E. to look at a problem like this?

Thank you.

 

[tiny-tim]

welcome to pf!

hi whiskeySierra! welcome to pf!

I know that maximum height along the path would occur where PE has the highest value along the entire path. KE would never be zero because the projectile never stops moving, only slows a little at maximum height.

So how (if possible) can I use W.E. to look at a problem like this?

ah, but you do know the value of KE at maximum PE …

because you know that the horizontal component of velocity is constant, and that will be the minimum KE (instead of 0)

 

[whiskeySierra]

Could we not say this occurs when the vertical component of velocity is zero?

 

[gneill]

Could we not say this occurs when the vertical component of velocity is zero?

Yes. Energy and momentum conservation laws apply individually and collectively to the vertical and horizontal components of the motion. If you have the initial speed and the launch angle you can immediately determine the initial horizontal and vertical components of the motion including velocity, momentum, and energy. Energy conservation (KE and PE) readily yields the maximum height of the projectile.

 

[asteriatic]

Great question! Yes, it is possible to use Work-Energy to analyze projectile motion. In fact, Work-Energy and Kinematics are closely related concepts and can be used interchangeably in many cases.

To find the maximum height of a projectile using Work-Energy, we can use the principle of conservation of mechanical energy. This states that the total mechanical energy (the sum of kinetic and potential energy) of a system remains constant, as long as there is no external work being done on the system.

In the case of a projectile with no air resistance, the only external force acting on the system is gravity. Therefore, the total mechanical energy of the projectile will remain constant throughout its motion.

At the maximum height, the projectile's velocity will be zero and all of its initial kinetic energy will be converted into potential energy. This means that at the maximum height, the total mechanical energy will be equal to the potential energy.

Using this principle, we can set the initial total mechanical energy (which is equal to the initial kinetic energy) equal to the potential energy at the maximum height, and solve for the maximum height using the formula for potential energy.

Alternatively, we can also use the work-energy theorem, which states that the net work done on an object is equal to the change in its kinetic energy. In the case of a projectile, the net work done on it is equal to its change in kinetic energy from its initial velocity to its final velocity (which is zero at maximum height).

Using this theorem, we can set the net work done on the projectile equal to the change in its kinetic energy and solve for the maximum height.

In summary, both Kinematics and Work-Energy can be used to analyze projectile motion. It is important to understand the principles and equations behind both concepts in order to choose the most appropriate approach for a given problem. Keep exploring and asking questions, that's what science is all about!