Mathematics

Grade Levels: 6th Grade, 7th Grade, 8th Grade, 9th Grade, 10th Grade, 11th Grade, 12th Grade,

Topics: AlgebraExpressions and Equations

Common Core State Standard: A-REI.2,

Knowledge and Skills:

• Can evaluate expressions by substituting values for variables

• Can simplify expressions using correct order of operations

• Can do basic operations on both sides of an equation in such a way as to preserve the equality

Lesson:

*Answers:*

1.

h (meters) |
v (meters/sec) |

15 | 26.2 |

40 | 14.0 |

35 | 17.1 |

5 | 29.7 |

30 | 19.8 |

50 | 0.0 |

25 | 22.1 |

20 | 24.2 |

45 | 9.9 |

10 | 28.0 |

0 | 31.3 |

2. 31.3 meters/sec

3. No. For the 50 foot roller coaster, the increase in velocity in the first half of the fall (from *h* = 50 feet to *h* = 25 feet) is 22.1 meters/sec. The increase in velocity in the second half of the fall (from 25 feet to 50 feet) is only 9.2 meters/second (31.3 – 22.1). So the velocity does not double when the length of the drop doubles.

**The Drop**

Suppose a roller coaster ride begins by climbing to a height of 50 meters, stopping briefly, and then falling rapidly to ground level (a height of 0 meters).

As it falls the roller coaster will gain speed.

If you ignore the effects of friction, then height and speed are related by this equation:

50 – *h* = v²/19.6

(*h* = height in meters, *v* = velocity in meters per second).

1. Find the missing values in the table below:

h (meters) |
v (meters/sec) |

15 | ? |

40 | ? |

? | 17.1 |

5 | ? |

? | 19.8 |

? | 0.0 |

25 | 22.1 |

? | 24.2 |

? | 9.9 |

? | 28.0 |

0 | ? |

2.

a) What is the velocity halfway to the bottom of “The Drop”, where h = 25?

b) What is the velocity at the bottom of the drop, where h = 0?

3. If you made the roller coaster twice as high, do you think the velocity at the bottom be twice as great? Explain your answer