IGCSE Physics- Gravitational Potential Energy = Kinetic Energy????

Did you ever wonder is there any relationship between the Gravitational Potential Energy and Kinetic Energy that you had learned in physics during your school days?

Some background knowledge:
Speed-time graph
Time to read this article: 5 ~ 7 minutes

In this blog we are going to explore this topic

First let us recap

Gravitational Potential Energy(g.p.e.) is:

m = mass in kg
h = height of the object above the ground(in meter)

g = acceleration of free fall:

Kinetic Energy(k.e.) is:

m = mass in kg
v = speed in m/s

Question:

Let us take a scenario for a ball,with initial 0 m/s velocity and free fall from a building. The ball took 5 seconds to reach the ground, and the ball has a mass of 0.5 kg, let's assume the air resistance against the ball are unnoticeable. Find the g.p.e

Finding g.p.e

Step 1: draw speed-time graph

If we want to find the g.p.e we need to know the total distance traveled by the ball in 5 seconds.

In simplicity we need to draw a speed-time graph, with the value of acceleration of free fall, as the following
at time t = 0, v = 0   m/s
at time t = 1, v = 10 m/s
at time t = 2, v = 20 m/s
at time t = 5, v = 50 m/s
The speed-time graph:

Step 2: find height,which is area under the curve from speed-time graph

Total distance is the area under the curve of speed-time graph
the final speed is 50 m/s , and the total time taken to reach the final speed is 5 seconds

by using the g.p.e ,

Finding k.e.

From the speed-time graph, we can also get the final velocity of the ball when the ball reach the ground which is 50 m/s

alternatively, since we know the velocity of the ball, we can now try to use the k.e. formulae to calculate its kinetic energy

Now you see, both computed  k.e.  and g.p.e are the same.

Math work out

let us use mathematical method to prove it

Eq1:

Step 1: Find formulae/representation to compute h

refer to the speed time graph, h is the area under the curve
Eq2:

v = final velocity at time t, which is 50m/s
t = total time taken to reach final velocity, which is 5 seconds

Step 2: Find formulae/representation to compute g

refer to the speed time graph,  g (acceleration) is  the gradient of that line
Eq3:

from the speed time graph, we know that v0 = 0, and v1 = 50 m/s and it took a total of t = 5 seconds
so as to compare with the v in the Eq2 , we can equate them as
Eq4:

by substituting Eq4 into Eq3 we get Eq5:

Step 3: Prove left hand side Eq = right hand side Eq

Finally we substitute Eq5 and Eq2 into Eq1:

then we just need to simplify the left hand side equation:

the left hand side equation is equivalent to the right hand side equation

We know that the kinetic energy is a scalar quantity. It does not contain direction, however, the g.p.e is indicating force pointing downwards.

We can conclude that the kinetic energy in the context of force of gravity ,it is the absolute/magnitude representation of the g.p.e


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