In science one year, we did a test of dropping a water bottle from different heights over a Pringle, and we had to protect the Pringle with a paper. But how would increasing the height increase the force is the mass and acceleration is the same?
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F=ma is the instantaneous force being exerted on an object.
So if you pick something up, and hold it in your hand, the weight you’re feeling in your palm is F=ma (in this case a being gravity, or g)
Now how high you hold the object, adds a third component, height. Which gives you a new equation.
Fp=mgh (Fp or PE) this is the potential energy that the object has, picking something up adds potential energy to the object, the higher you pick it up, the more energy, hence why it takes more energy to pick something up, or walk up stairs, than it does to just stand still or walk forward. Since m and g are constant, if you want to increase h, you need to add energy to the object in some way.
So this means if you had 2 equal sized balls on the ground, then picked up one in each hand, lifted the one in your left hand to a meter, and the one in your right hand to 2 meters, the object in your right hand would be storing twice as much potential energy. So when you drop them, you can expect the ball on the right to hit the ground with double the amount of force.
And that’s because as soon as you let go of the ball, it’s potential energy starts becoming kinetic energy. If we assumed both balls weigh 1kg, then the left ball should hit the ground with 9.81J of force, and the ball on the right would hit with 19.62J of force.
And we can prove this with the Kinect energy formula, KE=0.5mv^2, now this part gets trick, because we need to figure out the velocity that both object will have when they hit the ground. But we have their heights and accelerations. So that’s to Torricelli, we have a convenient formula that the velocity of an object falling from rest is v^2=2g(h) so after falling 1 meter it would be 4.429m/s. And this is where we see why the energy scale in a linear relationship with height. Most people know that with KE, the energy is proportional with an increase in mass, but squared to its velocity, so a an object with double the speed with have much more force than an object with double the weight. But with a falling object, while it’s speed increases linearity with time, it’s increase in velocity actually decreases every meter. Which might not be intuitive at first, but it’s because the fast an object travels, the less time it spends accelerating. So while it take 1 meter to accelerate to 4.429m/s and object needs to for 4meters to have double the speed, and then 16m to double again, for every doubling of speed, you need to square the distance an object falls.
What this means, is we have a directly linear relationship between the height of an object, and the kinetic energy it gains while falling. So double the height, double the energy, because even though the velocity has a greater impact, the increase in velocity per unit of height offsets it exactly.
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