What do you need to know about Free-fall, Projectile Motion, Air Resistance, and Terminal Velocity?

01 Jul What do you need to know about Free-fall, Projectile Motion, Air Resistance, and Terminal Velocity?

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Free-fall:  a special case of uniformly accelerated motion

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When you drop a hammer, it falls to the ground.  While it is falling, the only force acting on it is gravity. (There’s also a little bit of air resistance, but for a hammer moving at relatively low speeds, we can ignore that).  We call this state of motion “free fall“.

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Galileo was the first to figure out that objects in free fall all accelerate at the same rate.

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He didn’t have the technology to measure this precisely, but he could drop objects of different weights from a great height and see that they hit the ground at the same time.  He also performed a “thought experiment” that convinced him that heavy objects must accelerate at the same rate as lighter ones (still ignoring air resistance).

He imagined a big rock and a little rock tied together with a short rope.

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• If heavy objects fall faster than light ones, then if we drop the pair of rocks we should see the big rock trying to fall faster, pulling on the rope, with the smaller rock lagging behind, slowing it down.  The two rocks should accelerate at a rate in between the rate that either one would accelerate on its own.

• But when we tie the rocks together, we turn them into a single object whose weight is greater than either rock on its own.  The two rocks should accelerate at a rate that is faster than either one would accelerate on its own!

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So which is it?  Do the tied rocks fall slower or faster than the big rock would fall by itself?

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Galileo recognized that the only way this conflict does not exist is if the light rock and the heavy rock both accelerate at the same rate, regardless of whether they are tied together or not.  Even a feather will fall at the same rate as a hammer, if there’s no air resistance.  Don’t believe it?  Watch!

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Feather and Hammer Drop on the Moon

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Bowling Ball and Feather in world’s largest vacuum chamber

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Heavy objects fall at the same rate as light ones.

We’ll learn WHY this is true in a later note. For now, we can say that physicists have determined with a fair degree of precision that for an object in free fall near the surface of the earth, the acceleration due to gravity, which we call g, is 9.8 (m/s) / s, or 9.8 m/s^2.

If you drop a rock, after 1 second it will be going 9.8 m/s. After 2 seconds it will be going 19.6 m/s. And so on.

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The value of g actually varies a little bit from place to place, with a maximum of 9.83 and a minimum of 9.76 m/s^2.  If you are doing AP Physics, you can round to 10.  The physics teachers I know use 9.8 or 9.81 m/s^2.

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Projectile Motion

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An object in free fall is called a projectile.  Some of the earliest important applications of physics involved projectiles and were studied carefully at military academies, where students learned to predict the paths of stones launched by catapults or trebuchets or cannons.

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Naturally no one wanted to just shoot cannonballs straight up in the air, so they needed to understand motion in 2 dimensions, not just one.  The cannonball has to go up in the air but it also has to travel horizontally.

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In the absence of air resistance, the horizontal motion is completely independent of the vertical motion.

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If you understand vector addition well you will see why this must be true. Take vector V and vector H, at right angles to one another.  Their resultant is C.  Make V larger or smaller.  Does the H-component of the resultant vector change at all?  Nope. Make H larger or smaller.  Does the V-component of the resultant change?  Nope.  They’re independent.

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This is pretty convenient!  It means that if we know the initial velocity vector (speed and direction), we can know everything about the path of the projectile.

We find the vertical velocity component and apply our equations for accelerated motion, using g for acceleration.  That will tell us how high the projectile goes, and how long it is in the air.

Knowing how long it is in the air, we can use the horizontal component of the velocity to find out how far it goes.  There’s no acceleration in the horizontal direction because there’s no force acting to change the horizontal speed.

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Imagine you have a cannon aimed so that you can shoot a cannonball exactly horizontally. You also rig up your cannon in such a way that another cannonball drops straight to the ground at the very instant you fire the horizontal cannonball.  Which cannonball will hit the ground first?

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They land at the same time.  Both balls are affected by gravity from the moment they are released by the cannon.  Gravity does not ignore the horizontal cannonball just because it is going really fast.  It pulls it down at exactly the same rate as the ball that was simply dropped.

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This is often called the monkey-and-hunter problem, because instead of cannons the story often involves a hunter firing horizontally at a monkey hanging from a tree.  The monkey let go at the same instant the gun is fired, but sadly the monkey’s fall puts him right in the path of the falling bullet.  You might find this pretty easy to understand once you have thought about the independence of horizontal and vertical movement.

But what if we make it more complicated?  Instead of aiming horizontally, the hunter has to aim upward because the monkey is high up in the tree.  The monkey still lets go the instant the gun is fired.  Does he still get hit?  Let’s see:

Monkey and Hunter Demonstration

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Air Resistance and Terminal Velocity

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Real projectiles launched on earth experience air resistance.

At low speeds, for objects with a high ratio of weight to surface area (like, not a feather or a sheet of paper), we can ignore air resistance.

At high speeds, even an aerodynamic bullet experiences a lot of air resistance.

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Air resistance is a fairly complicated force.  We aren’t going to try to analyze it in detail.  But we can make a couple of important observations about it.

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• First, we can say that adding air resistance is going to lessen a projectile’s maximum height.

That means it will be in the air for a shorter time, so it can’t go as far. And the air resistance is also going to cause the horizontal component of the velocity to decrease as time goes by. That shortens the range of the projectile even more.  The path of the projectile is no longer a parabola.  The descent is much steeper than the ascent.

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• Second, the force of air resistance varies with speed, a lot.

When you double the speed of a ball, the air resistance doesn’t just double, it goes up by a factor of perhaps 4 (there’s no simple way to predict the exact number).  This explains the existence of a phenomenon we call terminal velocity.

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If you jump out of an airplane, at first you fall with an acceleration of 9.8 m/s^2.  But as your speed increases, the air resistance increases, so your acceleration quickly becomes less than g.

Still, you are speeding up, but that means that still, the air resistance is increasing.  So your acceleration continues to decline.  Where does this stop? Before long you will reach a speed where the upward force of air resistance matches the downward pull of gravity.  You’ll be falling very fast at this point, but you won’t be accelerating anymore. You’ve reached terminal velocity.  For a human in the air, terminal velocity is about 120 miles per hour.  So you can’t rely on it to save you if your parachute doesn’t open.

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Terminal velocity depends a lot on the shape and weight of the object.

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When you open your parachute, you effectively change your shape, and your terminal velocity becomes much smaller, so you slow down until you reach the terminal velocity permitted by the chute.  This should be a speed that is low enough to allow you to land safely.

You can watch the effect of terminal velocity in this breathtaking video of Felix Baumgartner jumping from space.

He’s so high up at the beginning that he is in free fall, with no air resistance.  His speed increases very quickly.  As he falls, he encounters increasingly dense air.  His speed is so great that even this thin air eventually stops his acceleration.  Watch the numbers on the screen and listen to the mission control guy to see how fast he is going at that point.  It’s a lot faster than 120 mph!

Felix Baumgartner Jumps from Space (Fast forward to about 57:00 into the video for the good stuff)

Alan Eustace Outdoes Baumgartner!