Exploring Creation with Physics Module 2: One-Dimensional Motion Equations and Free Fall

In Module 2 we will develop three very important equation s that will allow us to describe many situations which occur in one-dimensional motion. Free Fall: Whenever any object is falling towards the earth without anything inhibiting its fall, we say that the object is in free fall.

Relating Velocity, Acceleration, and Time v t a = Formula for Acceleration: v = v final - v initial t = t final - t initial a = v final - v initial t final - t initial a = v final - v initial t final v final = v initial + at final v = v o + at

Question 1: A car traveling at 35 m/sec (about 78 mph) passes a police car which is moving at 23 m/sec (about 51 mph). The policeman does not have a radar gun, so he immediately starts to speed up in order to match the speeder's velocity. That way, he will be able to measure how fast the speeder is moving. If the police car can accelerate at a rate of 2.5 m/sec 2, how long will it take for the policeman to match the speeder's velocity?

Relating Velocity, Acceleration, and Displacement We need to be able to relate velocity and acceleration to something other than time. Displacement is one of the fundamental quantities that we an measure with regard to motion, to it is natural that we find a way to relate velocity and acceleration to displacement. V= X t t final - t initia l X V= X t

Relating Velocity, Acceleration, and Displacement V= X t This formula is gives us the average velocity. However, there is another way we can calculate average velocity. If the object we are studying starts off with an initial velocity of v o and ends up with a final velocity of v, the, according to the definition of average, we could calculate the average velocity this way : v avg = v + v o 2 X t 2 = V=

Relating Velocity, Acceleration, and Displacement t v + v o 2 = X t = v + v o X 2 a = v t v + v o t t = v + v o a

Relating Velocity, Acceleration, and Displacement t = v + v o 2 X t = v + v o a and v + v o X 2 = a v 2 = v o 2 + 2a X

Question 2: A car has a maximum acceleration of 11,500 miles/hour 2. If it starts from rest and accelerates as quickly as possible for 500 yards, what will its velocity be? (There are 1760 yards in a mile ).

Relating Displacement, Velocity, Acceleration, and Time We learned in Module #1 that the area under the line will be equal to the displacement.

Relating Displacement, Velocity, Acceleration, and Time Area 1 = lengthwidth Area 2 = ½widthlength Total Area = Area 1 + Area 2 = 66 m m = 121 m

Relating Displacement, Velocity, Acceleration, and Time Area 2 = ½tv Area 1 = v o t a = v t = a vt v t Area 2 = ½ t a t a t2t2

Relating Displacement, Velocity, Acceleration, and Time Area 1 = v o t Area 2 = ½ a t2t2 Total Area = Area 1 + Area 2 = v o t + ½ at 2 x

Question 3: A race car starts at rest and travels 1,321 ft (a quarter of a mile) in 11 seconds. What was the car's acceleration?

Using Our Equations for One-Dimensional Motion x t v = v t a = v = v o + at v 2 = v o 2 + 2a X = v o t + ½ at 2 x

Question 4: If a car can accelerate from rest to a velocity of 60.0 miles per hour in 10.0 seconds, what is its acceleration ?

Question 5: In order to take off, a certain plane needs to start from rest and achieve a velocity of 150 miles/hour before it reaches the end of the runway. If the plane's acceleration is 20,000 miles/hr 2, what is the minimum length needed for a runway?

Question 6: A biker can maintain a constant acceleration of m/sec 2. If the biker starts from rest, how far can she travel if she keeps that acceleration up for 5.0 minutes?

Quick Review Remember, to apply the things that we have learned to any situation, two conditions must exists. 1. All motion must occur in a straight line. We are talking about one-dimensional motion here. Things can only travel one straight direction or in precisely the opposite direction. No curves, no bends in the road, nothing. All motion has to e perfectly straight. 2. The acceleration of any body that we study must be constant. If the acceleration is not constant, then the last three equations that we derived are not applicable.

Free Fall The motion of an object when it is falling solely under the influence of gravity. Objects falling near the surface of the earth experience a constant acceleration of 9.8 m/sec 2 (32 feet/sec 2 ) straight down. The value of 9.8 m/sec 2 or 32 feet/sec 2 is called the acceleration due to gravity and is often given the abbreviation of “g”. g = 9.8 m/sec 2

A Ball in Free Fall A ball is being dropped from a cliff The image of the ball is shown at several equal time intervals during the fall. The ball starts out with zero initial velocity While the acceleration is constant, the velocity is increasing At each time interval, the acceleration due to gravity increases the velocity, so the ball travels further with each time interval.

Free Fall The acceleration due to gravity is independent of the nature of the object experiencing free fall, as long as the object has mass. Suppose you hold a rock in one hand and a feather in another. If you let them go at the same time, which would hit the ground first? Why?

Experiment 2.1 The Acceleration Due to Gravity Is the Same for All Objects Watch Experiment 2.1

Apollo 15 Demonstration in 1971: Hammer and Feather Dropped at the Same Time on the Moon

Air Resistance When an object falls through the air, there are several gaseous molecules (like nitrogen and oxygen) and atoms (like argon) that are in the object's way. In order to fall, the object must shove the gaseous molecules and atoms out of its way. Well, the molecules and atoms resist this movement, and thus the object must force its way through them. A heavy object is much better at doing this than a light object. Therefore, heavy objects fall faster than light objects not because their acceleration due to gravity is larger, but because they are not as strongly affected by air resistance as light objects are.

Reaction Time Have you ever wondered how fast you can react to something that happens? Suppose you are driving a car and suddenly see an obstruction in the road ahead. How long will it take you to recognize the obstruction and move your foot from the accelerator to the brake? The time it takes you to do this is called your reaction time. As you might imagine, reaction time is different for different people. Some people react very quickly, and they make good race car drivers, airplane pilots, and tennis players. Other people just can't react quickly, and such fast-paced professions are not for them.

Experiment 2.2 Determining a Person’s Reaction Time

Experiment 2.2 Determining a Person’s Reaction Time

Reaction Time Now that you have the average displacement of the ruler as it fell, we can determine the time it took for you to notice that the ruler was falling and grasp it with your thumb and forefinger. = v o t + ½ at 2 x 11 cm = (0) (980 cm/sec 2 ) t + 1/2 t2t2 For the example, I am going to use my displacement of 11 cm t 2 = 2 (11cm) 980 cm/sec 2 t = 2 (11cm) 980 cm/sec 2 =.15 sec

Question 7: How long will it take a rock to hit the ground if it is dropped from the Leaning Tower of Pisa (height = 54.6 m )?

A more detailed look at free fall When an object is thrown upward in the presence of gravity, the object will reach its maximum height when the object's velocity equals zero. In addition, when it returns to the height from which it was thrown, its velocity will be equal to and opposite of its initial velocity.

Experiment 2.3 Factors that Affect Air Resistance

What happened in Experiment 2.3? When dropped with book, the book and papers all fell at the same rate because they all experienced the same acceleration of 9.8 m/sec 2. When we got rid of the book, the pieces of paper did not fall at the same rate. The wadded up piece of paper fell the fastest. The folded piece was not quite as fast as the wadded up piece of paper. The flat piece of paper fell the slowest.

What can we make of these results? First we can say the shape of an object influences the extent to which air resistance affects it. Thus, the more compact the shape, the lower the effect of air resistance. So, the orientation of an object affects the surface area as well.

The more surface area that the object exposes to the atoms and molecules that it must shove out to of the way, the more the air resistance. This means the terminal velocity of an object gets lower as the amount of surface area it exposes to the atoms and molecules that it must shove out of the way increases. In the end, then, we know at least three things that affect the terminal velocity of an object: its weight, shape and orientation as it falls.

Why do Parachutes work?

Terminal Velocity