Unit 3 - Constant Acceleration Particle Model

Constant Acceleration Particle Model

When an object is moving at a constant speed, it usually looks like this:

This is showing no speeding up, slowing down, or change in direction while the object is moving.  For example, if you were to push a bowling ball with a broom, and let it roll, it would continue to roll at a constant speed in the forward direction.

When an object is accelerating, there is an increase in the speed of the object.  For example:

Acceleration:

A change in the velocity over the change in time.  The acceleration is measured in meters per second squared (m/s^2).  The unbalanced force causes an increase or decrease in speed, or a change in direction.  The acceleration of the object is essentially how fast it is getting faster.  On a graph, a steeper slope shows a greater acceleration.  

Instantaneous Velocity: 

The average velocity for the entirety of a position v.s. time graph  is equal to the instantaneous velocity of the object at the midpoint of the graph.  In order to find the instantaneous velocity for an object at 5 seconds, you would have to calculate the average velocity from seconds 4 and 6.  ( xfinal - xinitial / tfinal - tinitial )

FORMULAS

Slope of a line ....... x final = velocity * change in t + xinitial 

Displacement on an x v.s. t graph ....... change in x = x final - x initial

Velocity on an x v.s. t graph ........ v = change in x / change in t

Acceleration ....... a = change in v / change in t

Displacement ....... change in x = 1/2 acceleration * (change in t)^2 + velocity initial * time

X V.S. T GRAPH QUESTIONS

a) What do they show you about the movement of the object?
- It shows you what direction the object is moving in, and whether it is speeding up or slowing down.

b) What does the shape tell you about the velocity of the object?
The shape shows if it is speeding up or slowing down.  If it is a steep slope, then it has a high velocity, and if it isn't a steep slope, the object is moving with a slow velocity.

c) What's the y-intercept of a x v.s. t graph?
x0 (x knot)

d) How do you use this graph to find the displacement?
Take x final  - x initial to find the displacement

V V.S. T GRAPH QUESTIONS

a) What do these graphs show you about the movement of an object?
- If the object's velocity is going in the opposite direction of the origin, then it is speeding up.  Likewise, the closer the line gets to the origin, the slower the object is moving.
- Constant velocity is shown with a straight line.

b) How do you determine the object's position with this graph? 

- The object is moving forward if the velocity's line is moving north.
- The object is moving backwards in the velocity's line is moving south.

c) How do you calculate the displacement?
- To calculate the displacement, you take the area of the space that is between the line and the origin.

A V.S. T GRAPH QUESTIONS

a) What do these graphs show about the movement of an object?
It shows the rate of acceleration of the object, if acceleration is present.

b) How do you determine the position of the object on an acceleration time graph?
- On a v v.s. t graph, if the slope is positive, the object has a positive acceleration.  This means the acceleration line will be above the origin.  Likewise, if the object has a negative slope on the v v.s. t graph, then the object has a negative acceleration.  This means the acceleration line will be below the origin.  

EXAMPLES

a.

The object is moving in the positive direction, however it's position versus time graph is curved.  This shows the acceleration of the object.  It's velocity over time is increasing as time passes, therefore it is speeding up.  Since it is moving in the positive direction, its velocity is positive, as well as its acceleration.  It's acceleration is constant because it is continuously speeding up.

b.

The object is constantly slowing down, meaning it's velocity's line is moving towards the origin in the positive direction.  It's acceleration is positive and constant, because the object is above the origin.

c.

The object initially has a quick speed, however it slows down and eventually stops.  This is represented in the v v.s. t graph because the velocity's line is going towards the origin.  There is still acceleration, however it is just negative.  If the object's velocity is at a negative slope, the acceleration is underneath the origin.  

d.

The object is moving in the negative direction the whole time, but it is still accelerating.  The velocity is increasing over time, which is shown by it moving further away from the origin.  Since the velocity is at a negative slope, the acceleration is underneath the origin.

PRACTICE PROBLEMS

a) Construct a motion map for the object's velocity and acceleration:.

b) Describe the motion of this object.

The object is constantly accelerating in the positive direction.  It's speed continues to increase as it moves.  

c) Determine the instantaneous velocity for the object at t=2s.  Explain.

v = change in x / change in t

v = 18 - 4 / 3 - 1

v = 14/2

v = 7 m/s 

To find the instantaneous velocity of an object at a certain time, take the plot from one second above and one second below the time you want to solve for, and use the change in x over the change in t, meaning the final minus the initial.

d) Assume the initial velocity was 10 m/s.  Determine the acceleration of the object.

a = change in v / change in t

a = 10 - 7 / 3 - 1

a = 3/2

a = 1.5 m/s^2

e) Show a corresponding velocity time graph for the object.