- To investigate equations that are important in real life.
- To practice substitution into equations.
- To practice with equivalent expressions and factoring out the HCF.
Imagine that you are standing on the side of the road. You see a driver coming down the road next to you who appears to be distracted by texting on their phone. Instead of turning with the curve of the road the driver begins to drive towards a fruit cart stationed near the side of the road. You scream “Look out!” - what happens next?
It takes the driver a little bit of time to look up and realize what is going on, and the car keeps moving. The distance it moves in this time is called the Reaction Distance.
Once the driver realizes, they hit the brakes to try and stop before they hit the cart. The car starts slowing down at this point, but it's still moving! The distance it moves in this time is called the Braking Distance.
Putting these two together gives the Total Stopping Distance, which is how far the car travels from when you first said "Look out!" to when it comes to a full stop.
Answer the questions below, then determine if the person will be able to stop in time to avoid hitting the fruit cart.
If $R$R is the speed of the car in miles per hour, then
- Reaction Distance (in feet) = $1.1R$1.1R
- Braking Distance (in feet) = $0.0515R^2$0.0515R2
- Total Stopping Distance (in feet) = $1.1R+0.0515R^2$1.1R+0.0515R2
For each of the formulas above, look at the right hand side of the equal sign and answer the following questions:
- How many terms are there?
- Are there any constant terms?
- What are the coefficients in each term?
- Are there any like terms? Why or why not?
- Factor the expression for the Total Stopping Distance formula. What was the HCF? Use the factored version of this equation instead of the original to answer the rest of the questions.
- Find the Reaction Distance, Braking Distance, and Total Stopping Distance for each of the following speeds:
- $60$60 miles an hour
- $25$25 miles an hour
- $10$10 miles an hour
- For each of the speeds add together your answers for Reaction Distance and Braking Distance. Compare this to your answer for Total Stopping Distance. What do you notice?
- Assume the driver was $60$60 feet from the fruit cart when you caught their attention. Which of the three speeds mean the driver is able to stop before the fruit cart? Which ones mean they crash into the cart? Justify your answer with your calculations.
- Create a table of values for each of these equations and a range of speeds. What kinds of patterns do you notice?
- Estimate the fastest speed that the car could be travelling while still avoiding a crash.
What factors may increase or decrease the Total Stopping Distance from one driver to the next? How will this change the equation? Compare with a friend! Did they think of anything you didn’t?