A Mathematical Model of Motion PHYSICS

1 Graphing Motion in One Dimension Interpret graphs of position versus time for a moving object to determine the velocity of the object Describe in words the information presented in graphs and draw graphs from descriptions of motion Write equations that describe the position of an object moving at constant velocity

Parts of a Graph X-axis Y-axis All axes must be labeled with appropriate units, and values.

Position vs. Time The x-axis is always “time” The y-axis is always “position” The slope of the line indicates the velocity of the object. Slope = (y2-y1)/(x2-x1) d1-d0/t1-t0 Δd/Δt

Uniform Motion Uniform motion is defined as equal displacements occurring during successive equal time periods (sometimes called constant velocity) Straight lines on position-time graphs mean uniform motion.

Given below is a diagram of a ball rolling along a table Given below is a diagram of a ball rolling along a table. Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds.   Notice that the ball covers an equal distance between flashes. Let's assume this distance equals 20 cm and display the ball's behavior on a graph plotting its x-position versus time.

The slope of this line would equal 20 cm divided by 0 The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This represents the ball's average velocity as it moves across the table. Since the ball is moving in a positive direction its velocity is positive. That is, the ball's velocity is a vector quantity possessing both magnitude (200 cm/sec) and direction (positive).

Steepness of slope on Position-Time graph Slope is related to velocity Steep slope = higher velocity Shallow slope = less velocity

Different Position. Vs. Time graphs Accelerated Motion Uniform Motion Constant positive velocity (zero acceleration) Increasing positive velocity (positive acceleration) Constant negative velocity (zero acceleration) Decreasing negative velocity (positive acceleration)

Different Position. Vs. Time Changing slope means changing velocity!!!!!! Increasing negative slope = ?? Decreasing negative slope = ??

X t B A C A … Starts at home (origin) and goes forward slowly B … Not moving (position remains constant as time progresses) C … Turns around and goes in the other direction quickly, passing up home

During which intervals was he traveling in a positive direction? During which intervals was he traveling in a negative direction? During which interval was he resting in a negative location? During which interval was he resting in a positive location? During which two intervals did he travel at the same speed? A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec

Graphing w/ Acceleration x Graphing w/ Acceleration t A B C D A … Start from rest south of home; increase speed gradually B … Pass home; gradually slow to a stop (still moving north) C … Turn around; gradually speed back up again heading south D … Continue heading south; gradually slow to a stop near the starting point

Tangent Lines t On a position vs. time graph: SLOPE VELOCITY Positive x Tangent Lines t On a position vs. time graph: SLOPE VELOCITY Positive Negative Zero SLOPE SPEED Steep Fast Gentle Slow Flat Zero

Increasing & Decreasing t x Increasing Decreasing On a position vs. time graph: Increasing means moving forward (positive direction). Decreasing means moving backwards (negative direction).

Concavity On a position vs. time graph: x Concavity On a position vs. time graph: Concave up means positive acceleration. Concave down means negative acceleration.

Special Points t Inflection Pt. P, R x Q R P S Inflection Pt. P, R Change of concavity, change of acceleration Peak or Valley Q Turning point, change of positive velocity to negative Time Axis Intercept P, S Times when you are at “home”, or at origin

2 Graphing Velocity in One Dimension Determine, from a graph of velocity versus time, the velocity of an object at a specific time Interpret a v-t graph to find the time at which an object has a specific velocity Calculate the displacement of an object from the area under a v-t graph

Velocity vs. Time X-axis is the “time” Y-axis is the “velocity” Slope of the line = the acceleration

Different Velocity-time graphs

Different Velocity-time graphs

Velocity vs. Time Horizontal lines = constant velocity Sloped line = changing velocity Steeper = greater change in velocity per second Negative slope = deceleration

Acceleration vs. Time Time is on the x-axis Acceleration is on the y-axis Shows how acceleration changes over a period of time. Often a horizontal line.

All 3 Graphs t x v t a t

Real life Note how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the orange segments would be connected. In our class, however, we’ll only deal with constant acceleration. v t a t

Constant Rightward Velocity

Constant Leftward Velocity

Constant Rightward Acceleration

Constant Leftward Acceleration

Leftward Velocity with Rightward Acceleration

Graph Practice Try making all three graphs for the following scenario: 1. Newberry starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed. 2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose. 3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail. 4. Fleeing the scene of the crime, Newberry takes off again in the same direction, speeding up quickly. 5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.

Area Underneath v-t Graph If you calculate the area underneath a v-t graph, you would multiply height X width. Because height is actually velocity and width is actually time, area underneath the graph is equal to Velocity X time or V·t

Remember that Velocity = Δd Rearranging, we get Δd = velocity X Δt So….the area underneath a velocity-time graph is equal to the displacement during that time period.

Area “positive area” “negative area” t v t Note that, here, the areas are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this as well. t x

Velocity vs. Time The area under a velocity time graph indicates the displacement during that time period. Remember that the slope of the line indicates the acceleration. The smaller the time units the more “instantaneous” is the acceleration at that particular time. If velocity is not uniform, or is changing, the acceleration will be changing, and cannot be determined “for an instant”, so you can only find average acceleration

3 Acceleration Determine from the curves on a velocity-time graph both the constant and instantaneous acceleration Determine the sign of acceleration using a v-t graph and a motion diagram Calculate the velocity and the displacement of an object undergoing constant acceleration

Acceleration Like speed or velocity, acceleration is a rate of change, defined as the rate of change of velocity Average Acceleration = change in velocity Elapsed time Units of acceleration?

Rearrangement of the equation

Finally… This equation is to be used to find (final) velocity of an accelerating object. You can use it if there is or is not a beginning velocity

Displacement under Constant Acceleration Remember that displacement under constant velocity was Δd = vt or d1 = d0 + vt With acceleration, there is no one single instantaneous v to use, but we could use an average velocity of an accelerating object.

Average velocity of an accelerating object would simply be ½ of sum of beginning and ending velocities Average velocity of an accelerating object V = ½ (v0 + v1)

So……. Key equation

Some other equations 2 This equation is to be used to find final position when there is an initial velocity, but velocity at time t1 is not known.

If no time is known, use this to find final position…. 2 2

Finding final velocity if no time is known… 2 2

The equations of importance

2 2 2 2 2

Practical Application Velocity/Position/Time equations Calculation of arrival times/schedules of aircraft/trains (including vectors) GPS technology (arrival time of signal/distance to satellite) Military targeting/delivery Calculation of Mass movement (glaciers/faults) Ultrasound (speed of sound) (babies/concrete/metals) Sonar (Sound Navigation and Ranging) Auto accident reconstruction Explosives (rate of burn/expansion rates/timing with det. cord)

4 Free Fall Recognize the meaning of the acceleration due to gravity Define the magnitude of the acceleration due to gravity as a positive quantity and determine the sign of the acceleration relative to the chosen coordinate system Use the motion equations to solve problems involving freely falling objects

Freefall Defined as the motion of an object if the only force acting on it is gravity. No friction, no air resistance, no drag

Acceleration Due to Gravity Galileo Galilei recognized about 400 years ago that, to understand the motion of falling objects, the effects of air or water would have to be ignored. As a result, we will investigate falling, but only as a result of one force, gravity. Galileo Galilei 1564-1642

Galileo’s Ramps Because gravity causes objects to move very fast, and because the time-keepers available to Galileo were limited, Galileo used ramps with moveable bells to “slow down” falling objects for accurate timing.

Galileo’s Ramps To keep “accurate” time, Galileo used a water clock. For the measurement of time, he employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which he collected in a small glass during the time of each descent... the water thus collected was weighed, after each descent, on a very accurate balance; the difference and ratios of these weights gave us the differences and ratios of the times...

Displacements of Falling Objects Looking at his results, Galileo realized that a falling ( or rolling downhill) object would have displacements that increased as a function of the square of the time, or t2 Another way to look at it, the velocity of an object increased as a function of the square of time, multiplied by some constant.

Galileo also found that all objects, no matter what slope of ramp he rolled them down, and as long as the ramps were all the same height, would reach the bottom with the same velocity.

Galileo’s Finding Galileo found that, neglecting friction, all freely falling objects have the same acceleration.

Acceleration Due to Gravity Galileo calculated that all freely falling objects accelerate at a rate of 9.8 m/s2 This value, as an acceleration, is known as g

Acceleration Due to Gravity Because this value is an acceleration value, it can be used to calculate displacements or velocities using the acceleration equations learned earlier. Just substitute g for the a

Example problem A brick is dropped from a high building. What is it’s velocity after 4.0 sec.? How far does the brick fall during this time?