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Skill 1.C
Science Practice 1 is about creating representations that depict physical phenomena and skill 1.C has a focus on creating qualitative graphs that represent features of a model or the behavior of a physical system.
This skill will be 0% of the multiple-choice section of the AP Physics 1 exam and will be only part of 20-35% of the free response dedicated to all of science practice 1. The individual skills do not have a % breakdown provided.
How will you recognize when to use this skill?
How will you recognize when to use this skill?
You will be given a blank graph and asked to Sketch without being given a table with numerical data to plot (which is what we use in skill 1.B).
Here is an example prompt: "On the axes provided sketch a graph of the velocity of the block as a function of time."
How is this different than plotting data?
How is this different than plotting data?
Sketched graphs are meant to go from concept to general sketch, which is the opposite of plotting data, where we graph the data and see what trend there is and then discuss what that shows about the situation.
Like plotted data, you should still have axes labels (but units are not needed), a zero-reference point marked on the y-axis, and the then you can sketch what you know. Note that the graphs are often scaled with variables and multiples or fractions of those variables.
See the example format of a sketched graphs at right.
How should I approach the sketching a graph?
How should I approach the sketching a graph?
Start with this question: What are the simple, relevant statements you can assume about key points along the x axis?
Is there a moment where something is at rest (which would mean a value of zero for velocity, momentum, and kinetic energy), a turnaround position (v=0 and position will reach a maximum or minimum)?
If so, put a data point to represent those key points, often as y intercepts, x intercepts, maxima, or minima on the graph.Sometimes there may be a "limiting case" scenario where a value approaches a maximum or minimum value. In these cases that limit may define a vertical or horizontal asymptote. These are not frequently found in our course but can show up on occasion.
Decide what shape to use to connect your dots with this question: What is the main functional relationship between the variables being plotted?
Many graphs have a set of facts we can memorize (slope of a velocity graph is acceleration, for example) and those facts can tell you if the slope should be constant or increasing or decreasing by matching to the conditions of the scenario. (for example, freefall is going to always have constant acceleration downward, so no acceleration on the horizontal axis and vertical acceleration is constant and downward).
This can often be answered by identifying a Physics equation or deriving a relationship between the variables. Before being confident enough to start your sketch, make sure any other quantities in the expression are constant in your scenario and solve the expression for the value on the y axis.
Compare the equation you determined with the standard relationships at right.
linear and direct both have a constant slope connecting key points, but linear does not necessarily go through the origin, while direct relationships must.
Quadratic relationships will be part of a parabola (although they do not necessarily start at 0 or at the apex as shown). Whether the curve is up or down will depend on the sign of the coefficient in front of the squared variable but can also be reasoned based on the physical nature of the changes.
Square root curves will stretch a curve horizontally and have no asymptote (they continue to infinity).
Inverse and inverse square relationships both have asymptotes of 0 on both the y and y axes. If you need to sketch these you must make sure they do not actually touch the axes.
Constant is not shown, but would just be a horizontal line because the values have an independent relationship to each other.
Example 1: Freefall vertical velocity
Example 1: Freefall vertical velocity
A ball is thrown off the edge of a cliff with a velocity of v at an angle of 30 degrees above the horizontal. Assume frictional forces are negligible. The ball reaches its maximum height at time t before falling to the ground. Sketch the vertical velocity as a function of time from when the ball is released until the ball hits the ground.
Sketch your own graph and then check with the points that might be evaluated to see if you earn all the points.
Example 2: Reaching Terminal Velocity
Example 2: Reaching Terminal Velocity
A thrill-seeking skydiversteps out of a plane and gains speed until eventually reaching a maximum speed v due to air resistance at a time t. Sketch the vertical velocity of the skydiver from t = 0 to t =2t.
Sketch your own graph and then check with the points that might be evaluated to see if you earn all the points.
Example 3: Mass on a Vertical Spring
Example 3: Mass on a Vertical Spring
A mass is hung on a vertical spring and allowed to come to rest at an equilibrium position of x = 0. The mass is then pulled down to a position of A and released from rest. The mass moves up and down, returning to its original position at time t. Sketch the position of the mass as a function of time from when the mass is released until it completes 3 complete cycles.
Sketch your own graph and then check with the points that might be evaluated to see if you earn all the points.
Example 1 Key Features and Possible Points
Example 1 Key Features and Possible Points
From step 1 of considering key points in time:
From step 1 of considering key points in time:
The initial velocity is angled, so the vertical velocity will be the vertical component of v, or vsin(30), which is 1/2 v
The highest point of the trajectory is a turnaround position where the velocity goes from + to -, so the velocity should be zero at a time of t
The ending velocity needs to be greater in magnitude, but opposite in direction to the initial velocity because there is a greater distance than the rise for the ball to fall with the same acceleration.
From step 2 of considering the functional relationship between the variables on the axes:
This is a constant acceleration situation so kinematics equations can relate v and t. v_final = v_initial +at shows a linear relationship vetween velocity and time and since the acceleration is downward (due to gravity), constant, and defined by the slope of this graph, the slope of the graph should be negative and constant.
Possible scoring:
Possible scoring:
1 point: the y intercept should be 1/2 v
1 point: the x intercept is at time = t
1 point: the graph should be negative and linear and continue past 2t, ending at a negative value greater than the initial velocity.
Look in the image carousel below these explanations for an example of what that could look like.
Example2 Key Features and Possible Points
Example2 Key Features and Possible Points
From step 1 of considering key points in time:
From step 1 of considering key points in time:
The initial velocity is 0 because this is a drop out of an airplane
The prompt tells us the velocity reaches a speed of v at t, so that is a point to plot. Note that the prompt does not dictate which direction is taken to be positive (up or down), so this can be +v or -v.
This situation is describing terminal velocity, which is when the velocity stops changing because air resistance has grown to the point where it counters gravitational force resulting in zero acceleration. This means the final speed needs to also be v.
From step 2 of considering the functional relationship between the variables on the axes:
This is a situation beyond our class's goals to evaluate during the change, but we can say that without air resistance at first (freefall) the vertical acceleration should be approximately 10 downward and at time t the acceleration should be zero (terminal velocity). Since the slope of the graph is acceleration that means the slope of the graph should be steep in the downward direction at first and then flatten to zero.
Possible scoring:
Possible scoring:
1 point: the y intercept should be 0 since they started from rest (vertically)
1 point: the line should pass through + or - v at time t (could be positive or negative depending on which direction is taken to be positive).
1 point: the graph should leave zero steeply and level out to be flat from t to 2t.
Look in the image carousel below these explanations for an example of what that could look like.
Example 3 Key Features and Possible Points
Example 3 Key Features and Possible Points
From step 1 of considering key points in time:
From step 1 of considering key points in time:
The mass is released from the stretched (downward) position of x = A so the starting position should be -A.
The promt tells us the cycle repeats at time t, and mass spring oscillators have a constant repeat time, so the graph should return to -A at multiples of t
Since x = 0 is the equilibrium position and simple harmonic motion (defined in unit 7) is a symetrical process, the maxima and minima should be equal but opposite. To the peaks should be at +A if the valleys are at -A.
From step 2 of considering the functional relationship between the variables on the axes:
The equation for the position of a simple harmonic oscillator (which describes ideal spring oscillators) is part of unit 7, x = Acos(2πft), is a sinusoidal graph, which is symetrical. This tells us the maxima and minima should be offset by 1/2 a period and the line should be curvy.
Possible scoring:
Possible scoring:
1 point: the y intercept should be -A
1 point: the data line should return to the starting position at t and 2t and 3t.
1 point: the data line should max out at a positive value of the starting position halfway between the minima and be sinusoidal.
Look in the image carousel below these explanations for an example of what that could look like.
More Practice
More Practice
This is a great card sorting activity for connecting position (time) graphs and velocity (time) graphs to verbal descriptions. That is the reverse of this skill, but you can practice
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