Kinematics in One Dimension Chapter 2

Kinematics describes motion in terms of Kinema is Greek for “motion” Kinematics → Describing motion Kinematics describes motion in terms of position displacement velocity (average and instantaneous) acceleration Dynamics Describing how interactions can change motion Mechanics Kinematics and dynamics.

Position x is the location of the object initial position final position positive x direction x axis at 0° For one-dimensional motion, plus (+) and minus (-) are used to specify the vector direction. Plus (+) is at 0° along the positive x axis and minus (-) is at 180° along the negative x axis.

Displacement Δx is the change in position Change Δ always means (final value) – (initial value)

Displacement is +5 m Example: Displacement in the positive x direction Displacement is the change initial final Displacement is +5 m

Example: Displacement in the negative x direction initial final Displacement is the change Displacement is - 5 m

Velocity is the displacement in one second. Average velocity is the displacement divided by the elapsed time. v "bar" means average SI units for velocity: meters per second

Example 2 The World’s Fastest Jet-Engine Car Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. The driver makes two runs through the course, one in each direction, to adjust for wind effects. Determine the average velocity for each run. Vector directions are shown with +/- signs.

Ignore the textbook definition of speed In this course, speed will mean the magnitude of the velocity. SI units for speed: meters per second

Instantaneous velocity is the average velocity for a very short time interval. Average velocity is the average during some time interval. Instantaneous velocity is the velocity at a specific moment.

Average acceleration is the velocity change in one second. "bar" means average

Example 3 Increasing velocity: Determine the average acceleration. v and a are in the same direction so v increases

Acceleration is the velocity change each second. v and a are in the same direction so v increases

Acceleration is the velocity change each second v and a are in opposite directions so v decreases

Graphical representation of kinematic quantities Position graph with a constant slope

Position graph with three different slopes Velocity = Slope

Position graph with a variable slope Velocity = Slope

Velocity graph with a constant slope

Your success in physics critically depends on your ability to correctly distinguish and correctly use technical terms as the terms are defined in physics. velocity and acceleration are different physical quantities with different units velocity is position change each second acceleration is velocity change each second velocity is different from velocity change

Five kinematic variables 1. position (at time t), x 2. initial velocity (at time t=0), vo 3. acceleration (constant), a 4. final velocity (at time t), v 5. final time, t The kinematic equations relate these 5 variables for situations where the acceleration is constant and allow us to solve for two of these quantities that are unknown.

Five kinematic equations for constant acceleration cases Position equations Velocity equations Obtained from the other equations by eliminating t. Use these equations to solve for any 2 missing variables. You need to know 3 of the 5 variables to find the missing 2 ! Derivation of the kinematic equations is shown at the end of the slides.

Success Strategy [ Very important steps !! ] 1. Make a drawing. 2. Draw direction arrows. (displacement, velocity, acceleration) 3. Label the positive (+) and negative (-) directions. 4. Label all know quantities. 5. Use a table to organize known values with correct units. 6. Make sure you have three of the five kinematic quantities. 7. Select the appropriate equations and solve. There may be two possible answers because quadratic equations have 2 roots. For multi-segmented motion, remember that the final velocity of one segment is the initial velocity for the next segment.

x a v vo t Example: Find the final position x m s ? 8s same directions so v increases Find the final position x x a v vo t m s ? 8s

x a v vo t m s ? 8s

x a v vo t m s ? Example 6 Airplane take-off Find the final position x same directions so v increases Example 6 Airplane take-off Find the final position x x a v vo t m s ?

Example 6 Airplane take-off v vo t m s ?

Free fall -- motion only influenced by gravity In the absence of air resistance, all bodies at the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is small compared to the radius of the Earth, then the acceleration remains constant throughout the motion. This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity. The direction of this acceleration is downward. The magnitude of the acceleration is called "g".

air no air (vacuum) (not vacuum cleaner) air resistance influences falling y downward y axis positive upward negative downward

A stone is dropped from the top of a tall building A stone is dropped from the top of a tall building. What is the displacement y of the stone after 3 s of free fall? y a v vo t m s ? -9.8 m/s2 0 m/s 3 s

Vertical velocity always momentarily zero at the top . A coin is tossed upward with an initial speed of 5 m/s. How high does the coin go above its release point? Ignore air resistance. Only one value is given in the wording of this problem. You are expected to add two values by thinking. Vertical velocity always momentarily zero at the top . y a v vo t ? +5 m/s

y a v vo t ? -9.8 m/s2 0 m/s +5.00 m/s

The velocity changes, but does the acceleration change? Acceleration versus Velocity There are three parts to the motion of the coin. On the way up, the vector velocity is upward and the vector acceleration is downward so the velocity change is downward (v gets less upward). At the top, the vector velocity is momentarily zero and the vector acceleration is downward so the velocity change is downward (v gets more downward). On the way down, the vector velocity is downward and the acceleration vector is downward so the velocity change is downward (v gets more downward). The velocity changes, but does the acceleration change?

The End

Derivation of the kinematic equations

multiply both sides by t Deriving the kinematic equations ...... We assume the object is at the origin at time to = 0. and Velocity definition multiply both sides by t position equation Also, for constant acceleration cases

multiply both sides by t Acceleration definition multiply both sides by t Velocity equation

another position equation