# A Mathematical Model of Motion

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A Mathematical Model of Motion
Chapter 5 A Mathematical Model of Motion

5.1 Graphing Motion in One Dimension
Position-Time graphs The position x, is plotted against time t on a coordinate system. How long is an object at a given location? If the time was any finite amount, the object would be at the same position during that time and would therefore have no movement! An instant of time must therefore be zero.

Using a Graph to Find Out Where and When
At what time was the object at ………. What was the position of the object at …….. Graphing two or more objects Use pictorial and graphical representations to determine when the two objects are at the same position. From Graphs to Words and Back Again Use your knowledge of motion graphs to determine what the object is doing.

Using an equation to find out where and when
Uniform motion Equal displacements occur during successive time intervals. Recall slope = rise/run Using an equation to find out where and when v = ∆d/ ∆t = df-di / tf-ti d = do + vt

Homework: Chapter 5 practice problems 1 – 12.

5.2 Graphing Velocity in One Dimension
Determining Instantaneous Velocity Recall v = ∆d / ∆t The slope of the curve at a given time will determine the velocity at that point. Use the tangent to the curve for changing velocities. (DEMO)

Displacement From a Velocity-Time Graph
Velocity-Time graphs Having the velocity at any given time allows us to graph the velocity of any object versus time. Displacement From a Velocity-Time Graph Displacement is found from a velocity time graph by taking the area under the curve. V t

Homework: Chapter 5 practice problems 13 – 16.

5.3 Acceleration Determining Average Acceleration a = ∆v / ∆t
Recall acceleration is a change in velocity

Constant and Instantaneous Acceleration
Determined by taking the slope of a velocity – time graph. If the graph is not linear, you can take the slope of the tangent line to determine the instantaneous acceleration.

Positive and Negative Acceleration
The sign of the acceleration is determined by taking the sign of the slope of the velocity – time graph. (vf – vi) Acceleration when instantaneous velocity is zero?

Calculating Velocity from Acceleration
a = ∆v / ∆t v = vo + at

Displacement Under Constant Acceleration
Use the area under the curve of the velocity – time graph to find the displacement when the velocity is constant. v t

The total displacement is just the sum of the rectangle and the triangle.
d = vot + ½ (v-vo)t if the initial position is not zero d = do + ½ (v+vo)t This equation gives us the final position with a constant acceleration.

If the velocity and time are known the equation can be rearranged as
d = do + vot +1/2at2 If the time is unknown but the velocity, distance and acceleration need to be related, use this equation: v2 = vo2 + 2a(d-do)

Homework: Chapter 5 practice problems 17 – 22.

5.4 Free Fall Acceleration Due to Gravity
No matter what an object is made of or its mass, all objects fall at the same rate when friction is ignored. This is called the acceleration due to gravity (g) and its value is m/s2. This acceleration is always downward and may be negative depending on the coordinate system. In formal labs use 9.78 m/s2.

PSS Sketch the problem. Draw a vector diagram showing the motions.
List all the variables and known and unknown values. Use the chart to select an equation to relate the variables. Solve the problem. Check your answer and units.

Homework: Chapter 5 practice problems 31 – 33.