Kinematics AP Physics 1

Defining the important variables Kinematics is a way of describing the motion of objects without describing the causes. You can describe an object’s motion: In wordsMathematicallyPictoriallyGraphically No matter HOW we describe the motion, there are several KEY VARIABLES that we use. SymbolVariableUnits tTimes aAccelerationm/s/s x or yDisplacementm vovo Initial velocitym/s vFinal velocitym/s g or a g Acceleration due to gravity m/s/s

Reference Frames Any measurement of position, displacement, velocity or acceleration must be made with respect to a defined reference frame—this is first step in problem solution. Possible reference frames: Window with up = + or – Un-stretched net with up = + or - Stretched net with up = + or – Ground = not sufficient information Module 2-1

Average Speed and Velocity Average velocity is a vector, so it has magnitude and direction. In one dimension we use + or minus sign to indicate direction. Module 2-6

Instantaneous Velocity instantaneous velocity is defined as the average velocity over an infinitesimally short time interval. Module 2-9

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.

Kinematic #1

Example: A boat moves slowly out of a marina (so as to not leave a wake) with a speed of 1.50 m/s. As soon as it passes the breakwater, leaving the marina, it throttles up and accelerates at 2.40 m/s/s. What do I know? What do I want? v o = 1.50 m/sv = ? a = 2.40 m/s/s t = 5 s a) How fast is the boat moving after accelerating for 5 seconds? 13.5 m/s

Kinematic #2 b) How far did the boat travel during that time? 37.5 m

Kinematic #3 What do I know? What do I want? v o = 12 m/sx = ? a = -3.5 m/s/s V = 0 m/s Example: You are driving through town at 12 m/s when suddenly a ball rolls out in front of your car. You apply the brakes and begin decelerating at 3.5 m/s/s. How far do you travel before coming to a complete stop? m

Common Problems Students Have I don’t know which equation to choose!!! EquationMissing Variable x v t

Up and Down Motion For object that is thrown upward and returns to starting position: assumes up is positive velocity changes sign (direction) but acceleration does not Velocity at top is zero time up = time down Velocity returning to starting position = velocity when it was released but opposite sign Module 4-3

Acceleration due to Gravity Module 4-4

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.

“g” or a g – The Acceleration due to gravity The acceleration due to gravity is a special constant that exists in a VACUUM, meaning without air resistance. If an object is in FREE FALL, gravity will CHANGE an objects velocity by 9.8 m/s every second. The acceleration due to gravity: ALWAYS ACTS DOWNWARD IS ALWAYS CONSTANT near the surface of Earth

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? What do I know? What do I want? v oy = 0 m/sy = ? g = -9.8 m/s 2 y o =0 m t = 5.78 s Which variable is NOT given and NOT asked for? Final Velocity! m H =163.7m

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. What do I know? What do I want? v o = 0 m/sa = ? x = 2.5 m v = 43.5 m/s Which variable is NOT given and NOT asked for? TIME m/s/s

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? What do I know? What do I want? v o = 0 m/st = ? x = 35 m a = 2.00 m/s/s Which variable is NOT given and NOT asked for? Final Velocity 5.92 s

Graphical Analysis of Motion AP Physics 1

Slope – A basic graph model A basic model for understanding graphs in physics is SLOPE. Using the model - Look at the formula for velocity. Who gets to play the role of the slope? Who gets to play the role of the y-axis or the rise? Who get to play the role of the x-axis or the run? What does all the mean? It means that if your are given a Displacement vs. Time graph, to find the velocity of an object during specific time intervals simply find the slope. Velocity Displacement Time

Displacement vs. Time graph What is the velocity of the object from 0 seconds to 3 seconds? The velocity is the slope!

Example It is very important that you are able to look at a graph and explain it's motion in great detail. These graphs can be very conceptual. Look at the time interval t = 0 to t = 9 seconds. What does the slope do? It increases, the velocity is increasing Look at the time interval t = 9 to t = 11 seconds. What does the slope do? No slope. The velocity is ZERO. Look at the time interval t = 11 to t = 15 seconds. What does the slope do? The slope is constant and positive. The object is moving forwards at a constant velocity. Look at the time interval t = 15 to t = 17 seconds. What does the slope do? The slope is constant and negative. The object is moving backwards at a constant velocity.

Slope – A basic graph model Let’s look at another model Who gets to play the role of the slope? Who gets to play the role of the y-axis or the rise? Who get to play the role of the x-axis or the run? What does all the mean? It means that if your are given a Velocity vs. Time graph. To find the acceleration of an object during specific time intervals simply find the slope. Acceleration Velocity Time

Velocity vs. Time Graph What is the acceleration from 0 to 6s? What is the acceleration from 6 to 9s? You could say one of two things here: The object has a ZERO acceleration The object has a CONSTANT velocity What is the acceleration from 14 to 15s? A negative acceleration is sometimes called DECELERATION. In other words, the object is slowing down. An object can also have a negative acceleration if it is falling. In that case the object is speeding up. CONFUSING? Be careful and make sure you understand WHY the negative sign is there.

Area – the “other” basic graph model Another basic model for understanding graphs in physics is AREA. Let's try to algebraically make our formulas look like the one above. We'll start with our formula for velocity. Who gets to play the role of the base? Who gets to play the role of the height? What kind of graph is this? Who gets to play the role of the Area? A Velocity vs. Time graph ( velocity = y- axis & time = x-axis) Time Velocity Displacement

Example What is the displacement during the time interval t = 0 to t = 5 seconds? That happens to be the area! What is the displacement during the time interval t = 8 to t = 12 seconds? Once again...we have to find the area. During this time period we have a triangle AND a square. We must find the area of each section then ADD them together.

Area – the “other” basic graph model Let's use our new model again, but for our equation for acceleration. What does this mean? Who gets to play the role of the base? Who gets to play the role of the height? What kind of graph is this? Who gets to play the role of the Area? Time Acceleration An Acceleration vs. Time graph ( acceleration = y-axis & time = x-axis) The Velocity

Acceleration vs. Time Graph What is the velocity during the time interval t = 3 and t = 6 seconds? Find the Area!

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 t (s) x (m) v (m/s)a (m/s/s) slope = v slope = a area = x area = v

Comparing and Sketching graphs One of the more difficult applications of graphs in physics is when given a certain type of graph and asked to draw a different type of graph t (s) x (m) slope = v t (s) v (m/s) List 2 adjectives to describe the SLOPE or VELOCITY The slope is CONSTANT The slope is POSITIVE How could you translate what the SLOPE is doing on the graph ABOVE to the Y axis on the graph to the right?

Example t (s) x (m) t (s) v (m/s) 1 st line 2 nd line 3 rd line The slope is constant The slope is “+” The slope is “-” The slope is “0”

Example – Graph Matching t (s) v (m/s) t (s) a (m/s/s) t (s) a (m/s/s) t (s) a (m/s/s) What is the SLOPE(a) doing? The slope is increasing