Kinematics (1-d) Mr. Austin

Motion Kinematics is the classification and comparison of an objects motion. Three “rules” we will follow: The motion is in a straight line The cause of the motion is ignored (coming soon!) The objects considered is a particle (not for long!) Particles and particle like objects move uniformly Ex. A sled going down a hill ANTI Ex. A ball rolling down a hill

Position The location of the particle in space. Needs a mathematical description to be useful. We assign a number to represent the particles position on a coordinate grid. There needs to be a zero point to reference The positions to the left are negative The positions to the right are positive

Vectors (more to come) A vector is a mathematical representation of something that has: Size Direction A scalar is a mathematical representation of something that has only size, but no direction. Direction is represented mathematically using a variety of methods. Angles Unit vectors Algebraic signs

Displacement Displacement is the change in a particles position It is a vector quantity Has a size Has a direction SI unit of: meters (m) Mathematically displacement is:

Sample Problem What is the displacement of a car that starts at x = 5 meters and ends at x = -3 meters? What is the displacement of a car that starts at x = -10 meters and ends at x = -12 meters?

Challenge What is your displacement if you run one lap on a round 400m track?

Displacement vs. Distance Displacement is only concerned with the difference between the starting point and ending point. It is a vector. Distance is the total length an object covers. It is a scalar.

Sample Problem What is the distance and displacement, from position A (25m) to F (-55m), of the car?

Distance Displacement

Plotting an Objects Position with Time

Average Velocity The rate at which the position of an object changes with time It is a vector Has a magnitude Has a direction SI unit: meter/second (m/s) Mathematically: This is the slope of a position time graph

Sample Problem What is the average velocity if you run the length of football field (91.4 meters) in 20 seconds?

Challenge What is the average velocity if you circumnavigate the globe in 3 days? Always remember, velocity deals with the change in position from where an object started to where it ended with respect to time. If you end where you started the displacement is zero!

Average Speed The rate that a distance is covered relative to time It is a scalar. Unit: m/s Mathematically: Challenge: Can average speed and average velocity be the same? Can they be different?

Sample Problem A car pulls out of a driveway and goes 5 meters forward than reverses 3 meters. All of this happens in 8 seconds. What is the average speed and velocity of the car? Average Speed Average Velocity

Book Practice for Homework Page 29 #1 Page 30 #1, 2, 3, 5, 8

Riddle A man wishes to marry a wealthy kings daughter. The wealthy king, hoping to make a fool of the man, gives him a test. If the man passes, the king explains, he will be married to the daughter. The man is blindfolded and taken outside. There are 20 statues in a line. All the statues are black except for one which is white. The blindfolded man must find the white statue to marry the daughter. How does he find it?

Instantaneous Velocity Mr. Austin traveled from Garnet Valley High School’s parking lot to the Franklin Institute (24.2 miles) in 42 minutes. What was Mr. Austin’s average speed? Am I in trouble? Instantaneous velocity is the velocity of a particle at any given moment in time. Can be positive, negative, or zero.

Instantaneous Velocity, graph The instantaneous velocity is the slope of the line tangent to the x vs. t curve This would be the green line The light blue lines show that as  t gets smaller, they approach the green line

Average Speed vs Speed Average speed is the distance traveled divided by the time it takes to travel. Its is a scalar. Speed is simply the magnitude of instantaneous velocity. Strip the velocity of any direction information It is a scalar

Acceleration The change in velocity of an object. It is a vector Has a size Has a direction Unit: Average acceleration is represented mathematically as:

Viewing Acceleration

Instantaneous Acceleration -- graph The slope of the velocity- time graph is the acceleration The green line represents the instantaneous acceleration The blue line is the average acceleration

Acceleration Expressed in g’s When accelerations are large we express them as a multiple of “g” It is the acceleration due to gravity near the surface of the Earth A man starts from rest and is accelerated to the speed of sound (340.2 m/s) on a rocket sled. This occurs in.75 seconds. What is his acceleration in terms of g?

Constant Acceleration This is a special case that tends to simplify things. Constant, or mostly constant, acceleration occurs all the time. Car starting from rest when a light turns green Car braking at a light when a light turns red There are a set of equations that are used to describe this motion.

Constant Acceleration Problem A car starts from rest and accelerates uniformly to 23 m/s in 8 seconds. What distance did the car cover in this time?

Book Practice Page 31 #22, 24, 28.

Free Fall Acceleration This is a case of constant acceleration that occurs vertically. All things fall to the Earth with the same acceleration In the absence of air resistance, all things fall to the Earth with the same acceleration: This is invariant of the objects dimensions, density, weight etc. When using the kinematic equations we use a y = -g = m/s 2

Free Fall – an object dropped Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is a y = -g = m/s 2 v o = 0 a = -g

Free Fall – an object thrown downward a y = -g = m/s 2 Initial velocity  0 With upward being positive, initial velocity will be negative v o ≠ 0 a = -g

Free Fall -- object thrown upward Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a y = -g = m/s 2 everywhere in the motion v = 0 v o ≠ 0 a = -g

Thrown upward, cont. The motion may be symmetrical Then t up = t down Then v = -v o The motion may not be symmetrical Break the motion into various parts Generally up and down

Free Fall Example Initial velocity at A is upward (+) and acceleration is -g (-9.8 m/s 2 ) At B, the velocity is 0 and the acceleration is -g (-9.8 m/s 2 ) At C, the velocity has the same magnitude as at A, but is in the opposite direction The displacement is –50.0 m (it ends up 50.0 m below its starting point)

Vertical motion sample problem A ball is thrown upward with an initial velocity of 20 m/s. What is the max height the ball will reach? What will the velocity of the ball be half way to the maximum height? What will the velocity of the ball be half way down to the hand? What is the total time the ball is in the air?

Book Practice Page 32 # 43, 47, 51.

Graphical Look at Motion: displacement – time curve The slope of the curve is the velocity The curved line indicates the velocity is changing Therefore, there is an acceleration

Graphical Look at Motion: velocity – time curve The slope gives the acceleration The straight line indicates a constant acceleration

The zero slope indicates a constant acceleration Graphical Look at Motion: acceleration – time curve

Test Graphical Interpretations Match a given velocity graph with the corresponding acceleration graph

Time (s) v (m/s) Interpreting a Velocity vs. Time Graph The area under the curve is the objects displacement.

Interpreting a Velocity vs. Time Graph The area under the curve is the objects displacement. Time (s) v (m/s)

Interpreting a Velocity vs. Time Graph The area under the curve is the objects displacement. Time (s) v (m/s)

General Problem Solving Strategy Conceptualize Categorize Analyze Finalize

Problem Solving – Conceptualize Think about and understand the situation Make a quick drawing of the situation Gather the numerical information Include algebraic meanings of phrases Focus on the expected result Think about units Think about what a reasonable answer should be

Problem Solving – Categorize Simplify the problem Can you ignore air resistance? Model objects as particles Classify the type of problem Substitution Analysis Try to identify similar problems you have already solved What analysis model would be useful?

Problem Solving – Analyze Select the relevant equation(s) to apply Solve for the unknown variable Substitute appropriate numbers Calculate the results Include units Round the result to the appropriate number of significant figures

Problem Solving – Finalize Check your result Does it have the correct units? Does it agree with your conceptualized ideas? Look at limiting situations to be sure the results are reasonable Compare the result with those of similar problems

Problem Solving – Some Final Ideas When solving complex problems, you may need to identify sub-problems and apply the problem-solving strategy to each sub-part These steps can be a guide for solving problems in this course

Quest Practice P. 29 Q1, 5a-c, 8; P. 30 P1, 6, 15, 22, 28, 35, 38, 40, 41, 47, 52, 63, 64, 74