Falling objects

Aristotle’s world is based on empirical observations:

earthAir Stars, fire

Moving objects before Galileo.Aristotle’s views dominated the scene.

Aristotle observed that heavy objects fall faster than lighter objects and reasoned that the rate of fall is proportional to the weight.

Aristotle reasoned from his observation of everyday events. Drop a hammer and a feather from the same height at the same time. It is obvious which one will reach the ground first.

Drop two objects at the same time, both fairly heavy so that air resistance can be neglected. Aristotle would say that the heavier object should reach the ground long before the lighter one. Actually, they hit at about the same time. Prepare two identical metal spheres. Cut one in half.Drop the two halves and the whole sphere at the same time.They will reach the ground at the same time.

BUT…….

There are several differences between Galileo and Aristotle.1. Galileo analyzes simple problems, such as a ship moving.

(no structure of the Universe mumbo-jumbo…)2. Galileo, like Aristotle, observes reality,

and tries to understand how things work. However, Galileo makes experiments to prove his conjectures.

Aristotle jumps from observations to conclusions, but he does not perform tests to prove or disprove his ideas.

The key idea is that reality is very complicated; we have to work on simple systems to understand how things work.AND we need to confront our ideas to reality to make sure we did not come up with some fantasies.

Galileo proposed that a theory (e.g., rate of fall proportional to weight) should be used to make a definite prediction, which should then be tested. If the prediction fails, then the established theory is wrong.

So, why were Aristotle’s ideas so accepted?Because they made sense.

Why are Galileo’s ideas superior?Because they also made sense, but, most importantly, because experiments proved Aristotle’s ideas to be wrong,while not disproving Galileo’s ideas.

To the extent that friction can be ignored, all objects fall alike.

After many experiments, Galileo came to the conclusion that

Today, we know that near the surface of the Earth, objects fall with a constant acceleration vector, which we call g. The vector  g  points downward and has magnitude close to 10 m/s2.

What does this mean?It means that for every second an object is in mid-air, the speed of an object increases by 10 m/s when the object is moving downdecreases by 10 m/s when the object is moving up.

Speed = absolute value of velocityV = 20 m/s Speed = 20 m/sV = -20 m/s Speed = 20 m/s

Tracking a Falling Object

• Starting from rest, its velocity has increased to 10 m/s after the first second; to 20 m/s after 2 s; to 30 m/s after 3 s; etc.• 10 m/s > 20 MPH• 30 m/s ≈ 70 MPH!

• During the first second, its average velocity is 10 m/s ÷ 2 = 5 m/s, during which time it travels 5 m. In the first half second, it travels 1.25 m.

ExampleSuppose that a brick falls from the top of a building and that it takes 4 seconds to reach the ground. How fast will it be going when it hits?

It gains 10 m/s for every second that it falls,so it will be going 4 x 10 m/s or 40 m/s when it hits.

Fact: An object thrown upward reaches the highest point of its trajectory when its velocity becomes zero (for just an instant).

Why? Its initial upward velocity keeps decreasing because of g until it stops

Fast

V = 0 here

Slower

Really slow

g

ExampleSuppose that an object is thrown upwards at a speed of 30 m/s. How long will it take to reach the highest point of its trajectory? It loses 10 m/s every second. In 3 seconds it will lose 30 m/s and stop. So, the time to reach the highest point is 3 seconds.

Earlier we learned that near the surface of the Earth, objects fall with a constant acceleration vector, which we call gThe vector  g  points downward and has magnitude close to 10 m/s2.

What does this mean?It means that for every second an object is in mid-air, the speed of an object increases by 10 m/s when the object is moving downdecreases by 10 m/s when the object is moving up.

Vtop = 0

Example: ball thrown upwards

Fact: Many of these problems are symmetric!

Imagine watching a movie of an object thrown upward, losing 10 m/s of speed every second.

Now imagine that the movie is running backwards.

Now you see an object falling downward, gaining 10 m/s every second.

i.e., the upward trajectory is just the same as the downward one, but running backwards.

Fact: If an object is thrown upward, the time for it to come back down to the same level is twice the time it takes to reach its highest point.

Example An object is thrown upwards at a speed of 20 m/s. How many seconds will it take to come back down? The time to reach its highest point is 2 seconds --- the time needed to lose 20 m/s of speed. The downward trajectory is exactly the upward one played backwards and will take the same amount of time --- 2 seconds. Total time = 2 s + 2 s = 4 s.

An object is thrown upwards with a speed of 30 m/s.How fast will it be going when it comes back down? Since the downward trip is just the upward trip played backwards, it will be going 30 m/s downward.

Alternatively: it takes 3 seconds to get to the top of the trajectory.When on top, the object has zero velocity and starts falling.The problem is symmetric, so it takes 3 seconds to get down.In these 3 seconds, its velocity will increase and reach the final value of 3x10 = 30 m/s.Careful: the final velocity will point down….

Guess what happens to the velocity….

30 m/s -30 m/s

tup = 3 s

Vtop = 0

tdown = 3 s y

Projectile motion

Goal: we want to predict the trajectory of projectiles.Why is this useful?

Cannonballs!(very important problem in the 1600s..)

Example. Gun shoots. I need to calculate the range.

v

D

Typically, I know the muzzle speed v and the distance D.I need to find the angle

Why? Because I want to stay in the gene pool…

Since this is complicated, let’s simplify…

Object moves horizontally on table and then falls on the ground

Important things to remember• the horizontal component remains the same.• the initial vertical velocity is zero.• the final vertical velocity is non-zero and points down.

Why? Because the horizontal acceleration is zero.The vertical acceleration points down.

g

We also find that:The trajectory is parabolic.

In the more general case, this is what happens

Velocity vector is tangent to trajectory(given w/out proof)

Look at the velocity components…

g

v

D

The maximum distance D is obtained when = 450

Applet:http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/ProjectileMotion/jarapplet.html

For those who are curious… D = v2sin(2)/2g(no, I am not going to ask this in the test…)

Problem-solving strategy:To find the range of a projectile on level ground when the initial velocity components are given: Use the vertical motion to find how long the projectile is in the air. Use the horizontal velocity component and the time-in-air to find how far the projectile goes.

Example.A motorcycle rider hits a take-off ramp and flies into the air with a vertical velocity component of 30 m/s and a horizontal velocity component of 40 m/s. How far will the rider travel horizontally before coming back down?

First figure the time in the air. Initially, the rider gains altitude at the rate of 30 m/s. Gravity will subtract 10 m/s from this rate for every second in the air. Thus, after 3 s, all of the vertical velocity component will be gone and the rider will be at the top of the arc.

The trip back down is just the trip up in reverse, so it will take 3 s also. The rider will be in the air for a total time of 6 seconds.

Now figure how far the rider goes. Horizontally, she travels 40 meters every second. Thus, she travels (6 s) x (40 m/s) = 240 m.

Last “fun” fact: there are always two angles that lead to the same D….Use the applet…

Applet:http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/ProjectileMotion/jarapplet.html

A cannon shoots a shell up with an initial vertical velocity component of 500 m/s and an initial horizontal component of 400 m/s. Neglecting air resistance and the curvature of the Earth, how far away from the cannon will the shell hit? (A) 50,000 m. (B) 20,000 m. (C) 40,000 m.

Is this realistic?

http://www.battleship.org/html/Articles/IowaClass/Main.htm

USS Missouri16-inch guns,V ~820 m/sRange ~41,000 m

It is realistic….

So, how do you think an ICBM works?