Physics 115 2012 Final Review Calculus is about “rates of change”




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Physics 115 2012

Final Review

Calculus is about “rates of change”.

Calculus is about “rates of change”.

A TIME RATE is anything divided by time.

CHANGE is expressed by using the Greek letter, Delta, D.

The MEANING?

For example, if t = 2 seconds, using x(t) = kt3=(1)(2)3= 8 meters.

The derivative, however, tell us how our DISPLACEMENT (x) changes as a function of TIME (t). The rate at which Displacement changes is also called VELOCITY. Thus if we use our derivative we can find out how fast the object is traveling at t = 2 second. Since dx/dt = 3kt2=3(1)(2)2= 12 m/s

Derivative of a power function



Unit Vector Notation



Example

A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north.

Dot Products in Physics



Example

Suppose a person moves in a straight line from the lockers( at a position  x = 1.0 m) toward the physics lab(at a position x = 9.0 m). “To the right” is taken as positive, as shown below

Example



Instantaneous Velocity

Instantaneous velocity is a measure of an object’s displacement per unit time at a particular point in time.

Instantaneous Acceleration

Instantaneous velocity is a measure of an object’s velocity per unit time at a particular point in time.

What do the “signs”( + or -) mean?



The 3 Kinematic equations

There are 3 major kinematic equations than can be used to describe the motion in DETAIL. All are used when the acceleration is CONSTANT.

Kinematics for the VERTICAL Direction

All 3 kinematics can be used to analyze one dimensional motion in either the X direction OR the y direction.

Examples

A stone is dropped at rest from the top of a cliff. It is observed to hit the ground 5.78 s later. How high is the cliff?

Examples

A pitcher throws a fastball with a velocity of 43.5 m/s. It is determined that during the windup and delivery the ball covers a displacement of 2.5 meters. This is from the point behind the body when the ball is at rest to the point of release. Calculate the acceleration during his throwing motion.

Examples

How long does it take a car at rest to cross a 35.0 m intersection after the light turns green, if the acceleration of the car is a constant 2.00 m/s/s?

Examples

A car accelerates from 12.5 m/s to 25 m/s in 6.0 seconds. What was the acceleration?

Summary

There are 3 types of MOTION graphs
  • Displacement(position) vs. Time

  • Velocity vs. Time

  • Acceleration vs. Time

There are 2 basic graph models
  • Slope

  • Area



Summary



Comparing and Sketching graphs



Example



Example – Graph Matching



Free Body Diagrams

A pictorial representation of forces complete with labels.

Free Body Diagrams



New’s 1st Law and Equilibrium

Since the Fnet = 0, a system moving at a constant speed or at rest MUST be at EQUILIBRIUM.

TIPS for solving problems
  • Draw a FBD

  • Resolve anything into COMPONENTS

  • Write equations of equilibrium

  • Solve for unknowns



Example

10-kg box is being pulled across the table to the right at a constant speed with a force of 50N at an angle of 30 degrees above the horizontal.
  • Calculate the Force of Friction

  • Calculate the Normal Force



Springs – Hooke’s Law

One of the simplest type of simple harmonic motion is called Hooke's Law. This is primarily in reference to SPRINGS.

Hooke’s Law from a Graphical Point of View



Example

A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount?

Newton’s Second Law

The acceleration of an object is directly proportional to the NET FORCE and inversely proportional to the mass.

Newton’s 2nd Law

A 10-kg box is being pulled across the table to the right by a rope with an applied force of 50N. Calculate the acceleration of the box if a 12 N frictional force acts upon it.

Example



Example (cont.)



Where does the calculus fit in?



TWO types of Friction

  • Static – Friction that keeps an object at rest and prevents it from moving

  • Kinetic – Friction that acts during motion



Force of Friction

  • The Force of Friction is directly related to the Normal Force.



Example

A 1500 N crate is being pushed across a level floor at a constant speed by a force F of 600 N at an angle of 20° below the horizontal as shown in the figure.

Example

If the 600 N force is instead pulling the block at an angle of 20° above the horizontal as shown in the figure, what will be the acceleration of the crate. Assume that the coefficient of friction is the same as found in (a)

Inclines



Example



Example



Horizontally Launched Projectiles

To analyze a projectile in 2 dimensions we need 2 equations. One for the “x” direction and one for the “y” direction. And for this we use kinematic #2.

Horizontally Launched Projectiles

Example: A plane traveling with a horizontal velocity of 100 m/s is 500 m above the ground. At some point the pilot decides to drop some supplies to designated target below. (a) How long is the drop in the air? (b) How far away from point where it was launched will it land?

Vertically Launched Projectiles

There are several things you must consider when doing these types of projectiles besides using components. If it begins and ends at ground level, the “y” displacement is ZERO: y = 0

Vertically Launched Projectiles

You will still use kinematic #2, but YOU MUST use COMPONENTS in the equation.

Example

A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees.

(a) How long is the ball in the air?

(b) How far away does it land?

(c) How high does it travel?

Example

A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees.

(a) How long is the ball in the air?

Example

A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees.

(b) How far away does it land?

Example

A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees.

(c) How high does it travel?

CUT YOUR TIME IN HALF!

A special case…

What if the projectile was launched from the ground at an angle and did not land at the same level height from where it started? In other words, what if you have a situation where the “y-displacement” DOES NOT equal zero?

Circular Motion and New’s 2nd Law

Recall that according to Newton’s Second Law, the acceleration is directly proportional to the Force. If this is true:

Examples



Examples

The maximum tension that a 0.50 m string can tolerate is 14 N. A 0.25-kg ball attached to this string is being whirled in a vertical circle. What is the maximum speed the ball can have (a) the top of the circle, (b)at the bottom of the circle?

Examples



Example



Example cont’



Momentum is conserved!

The Law of Conservation of Momentum: “In the absence of an unbalanced external force, the total momentum before the collision is equal to the total momentum after the collision.”

Several Types of collisions

Sometimes objects stick together or blow apart. In this case, momentum is ALWAYS conserved.

Work



Example



Example cont’

What if we had done this in UNIT VECTOR notation?

Example cont’



Elastic Potential Energy

The graph of F vs.x for a spring that is IDEAL in nature will always produce a line with a positive linear slope. Thus the area under the line will always be represented as a triangle.

Elastic potential energy



Energy is CONSERVED!



Example

A 2.0 m pendulum is released from rest when the support string is at an angle of 25 degrees with the vertical. What is the speed of the bob at the bottom of the string?


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