Graphing Motion

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 when the velocity of an object does not change 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) 2-5 s, 6-7 s B) 9-11 s C) 0-2 s D)7-9 s E) 6-7 s and 9-11 s, also 0-2 s and 5-6 s and 7-9 s

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.

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

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.