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physics.fisikastudycenter.com - Learning mechanical energy, the coservation of energy, the ratio of gravitational potential energy and kinetic energy will be discussed normally and the short-cut.

Problem
A ball falls from 20 m above the ground.


If the acceleration due to gravity g = 10 m/s² and the ball starts from rest, find the ratio of gravitational potential energy and kinetic energy of the ball when it reaches a distance of 15 m from its initial position!

Discussion

Calculation Steps
Step 1 : Find the distances and you should know the distances with the ground as a reference .
Take a look the figure above, the initial position is 20 m above the ground and when the ball reaches a distance of 15 m so then the remain distance is 5 m before reaching the ground.
Step 2 : Use the conservation of mechanical energy to get the ball speed.
We need two points (positions), you should take the initial position of the ball as the first point (h₁ = 20 m and the speed v₁ = 0 m/s), the second take the point asked when its reaches 5 m above the ground (h₂ = 5 m and unknown v₂).
Step 3 : Comparing Potential Energy (PE) and kinetic energy (KE).

Let's start!

1/2 mv₁² + mgh₁ = 1/2 mv₂² + mgh₂

by cancelling m then we get

1/2 v₁² + gh₁ = 1/2 v₂² + gh₂

inserting your data

1/2 (0)² + gh₁ = 1/2 v₂² + gh₂

v₂ = √[2g(h₁ − h₂)] = √[2gΔh] It's a well known equation.

v₂ = √[2(10)(20−5) = √300 m/s

Comparing PE and KE

PE : KE = mgh₂ : 1/2 mv₂²

by cancelling m

PE : KE = gh₂ : 1/2 v₂² = (10)(5) : 1/2 ( √300)² = 50 : 150 = 1 : 3

The short-cut
15 m represents the Kinetic Energy, 5 m represents the gravitational potential energy (PE) and 20 m represents the mechanical energy. Remember that the mechanical energy = KE + PE, similarly 20 m = 15 m + 5 m.

Caution : it doesn't mean that the mechanical energy is 20 m, the energy must be in joule not in meter!!

So then easily we get

PE : KE = 5 : 15 = 1 : 3

Try these!
From above question, now find the ratio of
1) PE : ME
2) KE : ME

(ME = Mechanical Energy)