•Motion

• Describing Motion.

• Change of position

• Passage of time

• Describe using a referent that is not moving.

• Change of position relative to some reference point (referent) over a defined period of time.

The motion of this windsurfer, and of other moving objects, can be described in terms of the distance covered during a certain time period.

• Measuring Motion.

• Speed.

– How much position has changed (displacement)

– What period of time is involved

– Speed = distance/time

– Constant speed = moving equal distances in equal time periods.

– Average speed = average over all speeds including increases and decreases in speed.

– Instantaneous speed = speed at any given instant.

• Considers a very short time period.

– v = d/t

This car is moving in a straight line over a distance of 1 mi each minute. The speed of the

car, therefore, is 60 mi each 60 min, or 60 mi/hr.

• Speed is defined as a ratio of the displacement, or the distance covered, in straight-line motion for the time elapsed during the change, or v = d/t.

• This ratio is the same as that found by calculating the slope when time is placed on the x-axis.

• The answer is the same as shown on this graph because both the speed and the slope are ratios of distance per unit of time.

• Thus you can find a speed by calculating the slope from the straight line, or "picture," of how fast distance changes with time.

If you know the value of any two of the three variables of distance, time, and speed, you can find the third.

What is the average speed of this car?

– Example

• if one travels 230 miles in 4 hours, then the speed is

• v = 230 mi. = 57.5mi/hr 4 hours

• Velocity.

– Velocity IS A vector quantity, it describes both direction and speed.

– A quantity without direction is called a scalar quantity.

Velocity is a vector that we can represent graphically with arrows. Here are three different velocities represented by three different arrows. The length of each arrow is proportional to the speed,

and the arrowhead shows the direction of travel.

• Acceleration.

– Three ways to change motion.

• change the speed.

• change direction.

• change both speed and direction at the same time.

– a change of velocity over a period of time is defined as acceleration (rate).

– acceleration = change of velocity time

(A) This graph shows how the speed changes per unit of time while driving at a constant 30 mi/hr in a straight line. As you

can see, the speed is constant, and for straight-line motion, the acceleration is 0.

(B) This graph shows the speed increasing to 50 mi/hr when moving in a straight line for 5 s. The acceleration, or change of

velocity per unit of time, can be calculated from either the equation for acceleration or by calculating the slope of the

straight line graph. Both will tell you how fast the motion is changing with time.

Four different ways (A-D) to accelerate a car.

– Example• if a car changes velocity from 55 km/hr to 80

km/hr in 5s, then• acceleration=80km/hr-55km/hr

5s• acceleration = 5km/hr/s• usually convert km/hr to m/s• then get m/s2

• represented mathematically, this relationship is:• a = Vf-Vi

t

• Aristotle's Theory of Motion.

• Aristotle thought that there were two spheres in which the universe was contained– The sphere of change– The sphere of perfection

• The Earth was seen to be a perfect sphere

– The Earth was fixed in place and everything else in the Universe revolved around the Earth

One of Aristotle's proofs that the Earth is a sphere. Aristotle argued that falling bodies move toward the center of the Earth. Only if the Earth is a sphere can that motion always be straight downward, perpendicular to the Earth's surface. Another of Aristotle's proofs that the Earth is a sphere (A) The Earth's shadow on the moon during lunar eclipses is always round. (B) If the Earth were a flat cylinder, there would be some eclipses in which the Earth would cast a flat shadow on the moon.

The shape of a sphere represented an early Greek idea of perfection. Aristotle's grand theory of the universe was made up of spheres, which explained the "natural motion"

of objects.

• Natural Motion

• Aristotle was the first to think quantitatively about the speeds involved in these movements. He made two quantitative assertions about how things fall (natural motion):

– Heavier things fall faster, the speed being proportional to the weight.

– The speed of fall of a given object depends inversely on the density of the medium it is falling through, so, for example, the same body will fall twice as fast through a medium of half the density.

• Notice that these rules have a certain elegance, an appealing quantitative simplicity.

– And, if you drop a stone and a piece of paper, it's clear that the heavier thing does fall faster, and a stone falling through water is definitely slowed down by the water, so the rules at first appear plausible.

– The surprising thing is, in view of Aristotle's painstaking observations of so many things, he didn't check out these rules in any serious way.

– It would not have taken long to find out if half a brick fell at half the speed of a whole brick, for example.

– Obviously, this was not something he considered important.

– From the second assertion above, he concluded that a vacuum cannot exist, because if it did, since it has zero density, all bodies would fall through it at infinite speed which is clearly nonsense.

• Forced motion

– For forced motion, Aristotle stated that the speed of the moving object was in direct proportion to the applied force. (Aristotle called forced motion, violent motion).

– This means first that if you stop pushing, the object stops moving.

– This certainly sounds like a reasonable rule for, say, pushing a box of books across a carpet, or a Grecian ox dragging a plough through a field.

• (This intuitively appealing picture, however, fails to take account of the large frictional force between the box and the carpet.

• If you put the box on a sled and pushed it across ice, it wouldn't stop when you stop pushing.

• Galileo realized the importance of friction in these situations.)

• Forces.

• A force is viewed as a push or a pull, something that changes the motion of an object.

• Forces can result from two kinds of interactions.

– Contact interactions.

– Interaction at a distance.

• The net force is the sum of all forces acting on an object.

– When two forces act on an object the forces are cumulative (the are added together.

– Net force is called a resultant and can be calculated using geometry.

• Four important aspects to forces.

– The tail of a force arrow is placed on the object that feels the force.

– The arrowhead points in the direction of the applied force.

– The length of the arrow is proportional to the magnitude of the applied force.

– The net force is the sum of all vector forces.

The rate of movement and the direction of movement of this ship are determined by a combination of direction and magnitude of force from each of the tugboats. A force is a vector, since it has direction as well as magnitude. Which direction are the two tugboats pushing? What evidence would indicate that one tugboat is pushing with greater magnitude of force? If the tugboat by the numbers is pushing with a greater force and the back tugboat is keeping the ship from moving, what will happen?

(A)When two parallel forces are acting on the cart in the same direction, the net force is the two forces added together.

• (B) When two forces are opposite and of equal magnitude, the net force is zero.

• (C) When two parallel forces are not of equal magnitude, the net force is the difference in the direction of the larger force.

• You can find the result of adding two vector forces that are not parallel by drawing thetwo force vectors to scale, then moving one so the tip of one is the tail of the other.

• A new arrow drawn to close the triangle will tell you the sum of the two individual forces.

(A) This shows the resultant of two equal 200 N acting at an angle of 90O, which gives a single resultant arrow proportional to a force of 280 N acting at 45O. (B) Two unequal forces acting at an angle of 60O give a single resultant of about 140 N.

• Horizontal Motion on Land.

• It would appear as though Aristotle's theory of motion was correct as objects do tend to stop moving when the force is removed.

– Aristotle thought that the natural tendency of objects was to be at rest.

– Objects remained at rest until a force acted on it to make it move.

• Aristotle and Galileo differed in how they viewed motion.

– Again, Aristotle thought that the natural tendency of objects was to be at rest.

– Galileo thought that it was every bit as natural for an object to be in motion.

• Inertia.

– Galileo explained the behavior of matter to stay in motion by inertia.

– Inertia is the tendency of an object to remain in motion in the absence of an unbalanced force such as:

• friction

• gravity.

Galileo (left) challenged the

Aristotelian view of motion and focused

attention on the concepts of distance, time, velocity, and

acceleration.

• Falling Objects

• Introduction

– The velocity of an object does not depend on its mass.

– Differences in the velocity of an object are related to air resistance.

– In the absence of air resistance (a vacuum) all objects fall at the same velocity.

– This is free fall which neglects air resistance and considers only the force of gravity on the object.

• Galileo vs. Aristotle's Natural Motion.

– Galileo observed that the velocity of an object in free fall increased with the time the object was in free fall.

– From this he reasoned that the velocity would have to be:

• proportional to the time of the fall

• proportional to the distance of the fall.

According to a widespread story, Galileo dropped two

objects with different weights from the Leaning Tower of Pisa. They were supposed to have hit the ground at about the same

time, discrediting Aristotle's view that the speed during the fall is proportional to weight.

(A) This ball is rolling to your left with no forces in thedirection of motion. The vector sum of the force of floor friction (Ffloor) and the force of air friction (Fair) result in a net force opposing the motion, so the ball slows to a stop.

(B) A force is applied to the moving ball, perhaps by a hand that moves along with the ball. The force applied (Fapplied) equals the vector sum of the forces opposing the motion, so the ball continues to move with a constant velocity.

– Remembering the equation for velocity:• v= d/ t

– We can rearrange this equation to incorporate acceleration, distance, and time.• rearranging this equation to solve for distance

gives.–d=vt

• An object in free fall should have uniform acceleration, so we can use the following equation to calculate the velocity.

–v= vf+vi 2

• Substitute this equation into d=vt, we get–d=(vf+vi) (t)

(2)

• The intial velocity is zero so:

–d=(vf) (t) (2)

• We can now substitute the acceleration equation in for velocity.

–a = vf-vi t

–vf=at

–d= (at) (t) (2)

• Simplifying we get

–d=1/2at2

– Galileo knew from this reasoning that a free falling object should cover the distance proportional to the square of the time of the fall.

– He also knew that the velocity increased at a constant rate.

– He also knew that a free falling object accelerated toward the surface of the Earth.

Galileo concluded that objects persist in their state of motion in the absence of an unbalanced force, a property of matter called inertia.

Thus, an object moving through space without any opposing friction (A) continues in a straight line path at a constant speed.

The application of an unbalanced force in the

direction of the change, as shown by

the large arrow, is needed to (B) slow

down, (C) speed up, or, (D) change the direction of travel.

• Acceleration Due to Gravity.

– Objects fall to the Earth with uniformly accelerated motion, caused by the force of gravity.

– All objects experience this constant acceleration.

– This acceleration is 9.8 m/s (32 ft/s) for each second of fall.

– This acceleration is 9.8 m / s2.

– This acceleration is the acceleration due to the force of gravity and is given the symbol g.

An object dropped from a tall building covers increasing distances with every successive second of falling. The distance covered is proportional to the square of the time falling (d t2).

The velocity of a falling object increases at a constant rate, 32 ft/s2.

– Example:

• A penny is dropped from the Eiffel Tower and hits the ground in 9.0 s. How far is if to the ground.

• d=1/2gt2

• d=1/2(9.8m/s2)(9.0s)2

• d=(4.9m/s2)(27.0s2)

• d= (ms2) s2

• d= m

• Compound Motion.

• Introduction.

– Three types of motion are:

• Horizontal, straight line motion.

• Vertical motion due to the force of gravity.

• Projectile motion, when an object is thrown into the air by a given force.

– Projectile motion can occur in several ways.• Straight up vertically.

• Straight out horizontally.

• At some angle in between these two.

– Compound motion requires an understanding of the following:

• Gravity acts on objects at all times, regardless of their position.

• Acceleration due to gravity is independent of the motion of the object.

• Vertical Projectiles.

– When an object is thrown straight upward, gravity acts on it during its entire climb.

– Eventually, the force of gravity captures the object and it begins to fall to Earth with uniformly accelerated motion.

– At the peak of the ascent, it comes to rest and begins its acceleration toward the Earth with velocity of zero.

On its way up, a vertical projectile such as a

misdirected golf ball is slowed by the force of

gravity until an instantaneous stop; then it

accelerates back to the surface, just as another golf

ball does when dropped from the same height. The

straight up and down moving golf ball has been moved to the side in the

sketch so we can see more clearly what is happening.

• Horizontal Projectiles.

– Horizontal projectiles are usually projected at some angle and can be broken down into horizontal and vertical components.

A horizontal projectile has the same horizontal velocity throughout the fall as it accelerates toward the surface, with the combined effect resulting in a curved path. Neglecting air resistance, an arrow shot horizontally will strike the ground at the same time as one dropped from the same height above the ground, as shown here by the increasing vertical velocity arrows.

A football is thrown at some angle to the horizon when it is passed downfield. Neglecting air resistance, the horizontal velocity is a constant, and the vertical velocity decreases, then increases, just as in the case if a vertical projectile. The combined motion produces a parabolic path. Contrary to statements by sportscasters about the abilities of certain professional quarterbacks, it is impossible to throw a football with "flat trajectory" because it begins to accelerate toward the surface as soon as it leaves the quarterback's hand.

Without a doubt, this baseball player is aware of the relationship between the projection angle and the maximum distance acquired for a given projection velocity.