As we once again join our intrepid feline travelers Cassie and Henri, we find them atop the Eiffel Tower (though worry not for today there will be no falling cats). Henri stares at the beautiful city below and sighs wishing for someone to share the experience with.
Unfortunately she’s with Cassie who at the moment is running around the platform chasing the brightly colored rubber ball she purchased earlier in the day. With a sad smile, Henri realizes that the rubber ball will probably be the highlight of the young kitten’s visit to Paris.
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Concluding that the view is too wonderful to waste on wistful thoughts, the older cat turns her attention back to the city below. However, it isn’t long before a kitten’s cry of despair fills the air. Rushing over, Henri finds Cassie leaning over the edge staring at the ground below. “Get away from there,” she cries, not wanting to repeat their experience at the Leaning Tower of Pisa (in that story cats did indeed fall). Obeying reluctantly, Cassie looks up at the older cat, her green kitten eyes moist.
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“But my ball bounced over the edge,” she sniffs. Henri stares out from the tower briefly then smiles.
“No need to be upset,” she answers gently patting the kitten’s head with her paw. “If you just wait your ball will come back to you.”
“Right,” scoffs Cassie (sure she’s a kitten, but a very rational one). “Since when do you believe in magic?”
Henri laughs. “Magic, oh no. I’m talking about Physics.”
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Not Physics again, thinks Cassie remembering their experience at the Leaning Tower of Pisa. However, it’s not like she has any other choice so she watches where the ball had bounced off the tower and waits.
Almost immediately, a surprised kitten sees her ball slowly come into view and stop before starting to fall again. Henri (who’s actually much faster than she looks) deftly reaches over and taps the ball back onto the tower into Cassie’s waiting paws.
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“Oh thank you, Physics!” exclaims the little kitten as she rolls onto her back holding the ball tightly in her paws and biting it.
Henri can’t help but smile at Cassie’s joyful reaction. However, it was getting late (and Henri was getting hungry) so they needed to be going. “Time to put the ball away. I know a place near the river that has the freshest fish.”
“I love fish,” cries the excited kitten as she puts the ball in her little pink backpack. The two of them set off and are soon enjoying a tasty fish dinner. Well at least until Cassie gets too close to the water and falls in, but that’s a story for another time.
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Now to be honest, in the real world (you know, the one where cats don’t travel around Europe and talk) the ball wouldn’t have bounced all the way back to the top of the tower. Yes, we have once again made use of that ideal world (which only exists when trying to explain scientific principles) where there are no energy losses (such as air resistance and material damping) and every bounce is true. With that in mind, let’s examine what happened with our improbable bouncing ball.
Well, now what’s the first thing you need for something to fall? Well height of course. Cassie, being the rambunctious kitten she is, provided that by carrying the ball in her little pink backpack as she raced up the stairs of the Eiffel Tower. Carrying the ball to the top of the tower is what physicists call Work.
The author's narrative has been misappropriated; report any instances of this story on Amazon.
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Well I guess everyone would call that work, but in physics Work is defined as applying a force to an object over some distance.
Now to carry any object you have to apply a force equal to its weight. As you may recall from our feline friends’ visit to the Leaning Tower of Pisa, an objects weight is its mass multiplied by gravitational acceleration.
So the distance has to be the length of the stairway Cassie ran up, right? Actually, no. Since gravity only acts vertically, the only distance we care about is the height of the tower. And it doesn’t matter how you get to the top. Henri, being older and a bit larger (once again don't call her fat), of course took the elevator. If she’d taken the ball with her, the Work done on the ball would be the same. The only difference is that the elevator would be doing the Work.
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Okay, so we’ve (well Cassie, actually) did Work on the ball. Big deal, you might say, what does that get us? Ah, by her simple act, Cassie has bestowed our mild-mannered ball with powers far beyond those of mortal...
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Sorry, got a bit carried away. Work is a form of energy and the total amount of energy in an object will always remain constant (in the absence of energy losses). Since the ball was moved against gravity, the Work done to it is stored in the ball as Gravitational Potential Energy. Since the amount of energy remains the same (in other words, energy is conserved), the Gravitational Potential Energy is equal to the amount of Work done to the ball.
So now the little kitten’s at the top of the tower playing with her ball, until (horrors!) it flies over the side. What happens? Why it falls of course and as it falls, its Gravitational Potential Energy is converted into yet another form of energy: Kinetic Energy. This is the energy an object has when it’s moving.
As the ball falls it contains both types of energy (which it finds very confusing), but the total amount of energy will always equal the initial Work done to the ball. Warning: the following picture contains mathematical equations. Read them quickly, it will hurt less.
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So what happens when the ball reaches the ground (h = 0). The ball’s Kinetic Energy will equal the initial Work done to the ball. Knowing this we can calculate the ball’s speed when it hits the ground. Now there was going to be a wonderful set of equations here, but Math Cat is asleep. Ah well.
Now knowing the upper platform of the Eiffel Tower is 900 feet high and that gravitational acceleration is 32.174 feet per second squared we find that the ball hits the ground at 240.6 feet per second or 164 miles per hour (splat!). Fortunately for Cassie, her ball is made of rubber so it merely flattens as it hits the ground.
Generally if something hits the ground it’s going to stop moving. So what happens to all that Kinetic Energy the ball just had? Is conservation of energy just another myth propagated by well-meaning parents? Nope, as the ball flattens against the ground, the Kinetic Energy is converted into Elastic Potential Energy (boy, who’d have thought a falling ball could be so complicated?).
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In the simplest sense, Elastic Potential Energy is generated when you compress or extend a spring (think pogo stick or Slinky) and our rubber ball is just another type of spring. The ball will reach a point of peak compression (it will look like a hockey puck) when all the Kinetic Energy has become Elastic Potential Energy.
So what happen next? Well, the same thing that happens if you push on a spring and let go, it shoots back to its original size. Now everything goes in reverse, as the ball rebounds, its Elastic Potential Energy turns into Kinetic Energy which becomes Gravitational Potential Energy as the ball flies upward.
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When the ball reaches the top of the tower, it stops for a brief instant (before falling again) and once again has its original Gravitational Potential Energy. Without Henri’s timely intervention, this back and forth journey would’ve continued forever (well in the ideal Physics world anyway).
Only one question remains, why was Henri so sure the ball would come back? Is it possible she’d already seen it even before reassuring Cassie? Let’s find out. At the Tower of Pisa we learned that acceleration is the measure of how much an object’s speed changes over time (a sports car going from 0 to 60 mph in 5 seconds for example). So the speed of the ball after some amount of time would be the gravitiational acceleration multiplied by time.Since we know the ball’s speed when it hits the ground, we can calculate how long to took to get there which is about 7.5 seconds.
The ball takes the same amount of time to go back up so it would reappear at the top of the tower every 15 seconds (no wonder Cassie didn’t have to wait long). I’ll bet clever Henri did indeed see it.
And so once again it's Physics to the rescue. Cassie was reunited with her ball and in the process might just be starting to appreciate Henri's fascination (some might say obsession) with Physics.