Chapter 5 A Mathematical Model Of Motion Answers

Graphing Motion in One Dimension

-Position Time Graphs

Used to analyze the motion of an object mathematically and to make predictions about its position, velocity, and acceleration

Example question using the graph above:

1. At what time was the position of the object equal to 40m?

Answer: 4 seconds

2. What was the position of the object at 2 seconds?

Answer: 20m

Graphing the Motion of Two or More Objects

The two lines (both red and blue) represent students. You can use the information in this position-time graph to determine both students positions.

Example question:

At what time do the students cross paths?

Answer: at approximately 2.2 seconds

Uniform Motion

Uniform motion : equal displacements occur during successive equal time intervals
A motion diagram and a position-time graph can be used to describe the uniform motion of the plane
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points
Using the first graph, we can find slope like this:

Graphing Velocity in One Dimension

-Determining Instantaneous Velocity

Instantaneous Velocity: velocity at a single point in time (slope of tan line)

-Velocity-Time Graphs

Average acceleration: the acceleration of an object between 2 points in time

Displacement: the linear distance traveled (the area under the curve)

-If the velocity is constant, the displacement is proportional to the width of the rectangle

-If you plot the displacement versus time, you will get a straight line with slope equal to velocity

-If the velocity is increasing, then the area of the rectangle increases in time, so the slope of a displacement-versus-time graph also increases

The shaded area is representative of the displacement during from 0 seconds to 6 seconds.

When you multiply the base and the height, we can see the displacement is 180 m.

*Area=(6 s) * (30 m/s)

=180 m

Acceleration

-Constant acceleration: a constant slope on a velocity-time graph

-Instantaneous acceleration: the acceleration of an object at an instant of time; the slope of the tangent to the curve on a velocity-time graph

*found the same way you find the slope of the tangent to the curve on the position-time graph

Positive and Negative Acceleration:

Positive acceleration is shown by a positive slope, or increasing

Negative acceleration is shown by a negative slope, or decreasing

Equations of Motion for Uniform Acceleration

Free Fall

-Acceleration due to gravity: the acceleration of an object in free fall that results from the influence of Earth's gravity

-ex. You drop a rock. A second later, it's velocity is 9.80 m/s downward. For each second that the rock is falling, its downward velocity increases by 9.80 m/s