Learning Targets
2.07B - I can use the constant velocity model to solve problems.
2.07C - I can use the constant velocity model to solve advanced problems that may require systems of equations.
2.08B - I can use the constant acceleration model to solve problems.
2.08C - I can use the constant acceleration model to solve advanced problems.
2.07C - I can use the constant velocity model to solve advanced problems that may require systems of equations.
2.08B - I can use the constant acceleration model to solve problems.
2.08C - I can use the constant acceleration model to solve advanced problems.
Can you...
- Use equations and/or graphs to solve word problems with constant velocity model.
- Use equations and/or graphs to solve word problems for a uniform acceleration model
Deriving and Using the Kinematic (Motion) Equations
Consider the velocity-time graph shown to the right. What kind of motion does it show? Because the slope is not zero and constant, it shows uniform acceleration. By avoiding values, this is a generic representation of an object accelerating. This means we can use it to derive equations. Try to derive the equations using the handout below. When you get stuck, scroll down for hints and video guides.
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STEP #1:
Does this graph show constant velocity or acceleration? Write “Constant Velocity” or “Acceleration” on your notes.
Label the following parts of the graph Vi, Vf, Δt, and ΔV |
STEP #2:
How would you calculate Δx from this graph?
Write an equation that will allow you to calculate Δx using ONLY the variables Vi, Δt, ΔV. Now use what you’ve done to create a NEW equation that allows you to calculate Δx using ONLY Vi, Δt, and a (acceleration). Box your final equation. |
STEP #3:
You have just created an equation for Δx using ONLY Vi, Δt, and a! Congratulations! But it looks like every equation we have to model this type of motion has a time included. What if we don’t know about the time? It might be nice to have an equation to model accelerated motion, which is not dependent on time…
What does the slope of a velocity-time graph tell us? Write an equation for the slope of a velocity-time graph using only the variables Vi, Vf, Δt, and a. Box this equation. |
STEP #4:
Using information from the previous questions, how could we rearrange the equation above so that we have a “new” equation for Δt using only Vi, Vf, and a?
Remember whatever we do to one side of the equation we must do to the other. |
STEP #5:
In Step #4 we found an equation for ∆ t . We also found the equation for ∆x in Step #2. Wherever you see a ∆ t in this equation, substitute the equation from Step #4.
This gives us an equation to model accelerated motion without any time variables!!! Now, simplify the equation above so each variable only appears one time. This is going to be challenging, but if you show you work one step at a time, I am confident you can succeed! |
Kinematic Equation Derivation: ALL STEPS
Example Problem Solving
These are some opportunities for practice. Please take the time to solve them using a mathematical solution, then check using a position vs. time graph, a velocity vs. time graph, or an acceleration vs. time graph.
You should be striving for ORGANIZED SOLUTIONS, not just answers. That means writing all 8 variables and filling in what you know before doing ANY math. When you do the math, you should write the equation you plan to use, then substitute in the values you know. |
STEPS FOR SHOWING WORK:
Step 1: Write out the variables and plug in what you know. Step 2: Write out the equation with NO NUMBERS Step 3: Rewrite the equation with numbers plugged in Step 4: Do MATH Step 5: Repeat as necessary |
Example 1A MAC Truck starts from rest and reaches a speed of 8.5 m/s in 20 seconds. What was the acceleration? How far did the truck travel?
Try to solve this problem with equations and when you are ready look at the video key to the left. Remember to follow the steps above for showing work. |
Example 2
A dune buggy travels for 20 seconds at a constant speed of 8.5 m/s. 30 seconds later, he came to a stop. What was his acceleration in the last 30 seconds? How far has the buggy travelled during the entire 50-second trip? (hint: You will need to find the change in position for each part of the trip)
Try to solve this problem with equations and when you are ready look at the video key to the left. Remember to follow the steps above for showing work. |
Example 3
A driver sees a deer in the road ahead and applies the brakes. The car slows to a stop from 8.5 m/s in 20 seconds. After 20 seconds, the deer moves and the car accelerates at a speed of 10 m/s/s for 3 seconds. How far has the car traveled since the driver saw the deer?
Try to solve this problem with equations and when you are ready look at the video key to the left. Remember to follow the steps above for showing work. |