What is the Relationship Between Potential and Kinetic Energy in Physics?

  • Context: Undergrad 
  • Thread starter Thread starter PhysicsPrac
  • Start date Start date
  • Tags Tags
    Calculus Physics
Click For Summary
SUMMARY

The relationship between potential energy and kinetic energy is defined by the equations Potential Energy = mgh and Kinetic Energy = 1/2mv^2. When equating these two forms of energy, the velocity v can be derived as v = sqrt(2gh). However, the discussion highlights a common misconception when substituting v with dx/dt and integrating, leading to the equation x = t*sqrt(2gh}. The confusion arises because h is not constant; it varies with time, which affects the application of the kinematic equation x = x0 + v0t + 1/2at^2.

PREREQUISITES
  • Understanding of classical mechanics concepts, specifically energy conservation.
  • Familiarity with calculus, particularly integration and differentiation.
  • Knowledge of kinematic equations in physics.
  • Basic understanding of gravitational potential energy and kinetic energy equations.
NEXT STEPS
  • Study the principles of energy conservation in classical mechanics.
  • Learn about the implications of variable height in potential energy calculations.
  • Explore advanced kinematic equations and their applications in varying conditions.
  • Investigate the relationship between velocity, acceleration, and time in dynamic systems.
USEFUL FOR

Students of physics, educators teaching mechanics, and anyone interested in understanding the dynamics of energy transformation in physical systems.

PhysicsPrac
Messages
6
Reaction score
0
Hi,

here is my question:

Potential energy = mgh
Kinetic energy = 1/2mv^2
mgh = 1/2mv^2, solving for v you get v = sqrt(2gh)

Now I know v is the same as dx/dt so if I substite in dx/dt for v :

dx/dt = sqrt(2gh)

Multiply by dt:
dx = sqrt(2gh) * dt
Integrate:
x = t*sqrt(2gh)

This should be right, but for some reason it doesn't work when I check it with the well known x = x0 + v0t + 1/2at^2 formula, what am I doing wrong?
 
Physics news on Phys.org
h is dependent on t, ie it's not a constant. What you calculated is if a ball fell a height h under gravity, and then started moving at constant velocity (at which point you say t=0) then the ball would move x = t*sqrt(2gh) in time t.
 
Oh I see, thanks for your help!
 

Similar threads

Undergrad Lagrangian approach for the inclined plane
  • · Replies 21 ·
Replies
21
Views
4K
Undergrad How can we prove the kinetic energy equation
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad What is the zero point for Potential Energy and how to find it from integral limits?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Kinetic Energy derivation assumption?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Car, gas and total energy calculation
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Calculus and Kinematic equations--- seeing the logic
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad The proportion of kinetic energy of a rotating rigid body?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Why isn't Kinetic Energy always equal to Potential Energy?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad System, potential energy, and nonconservative forces: The whole story
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Calculating Water Pool Depth: Factors to Consider in Fluid Dynamics
  • · Replies 13 ·
Replies
13
Views
2K