## Motion with Constant Acceleration

3.4 Motion with Constant Acceleration

### Learning Objectives

By the end of this section, you will be able to:

- Identify which equations of motion are to be used to solve for unknowns.
- Use appropriate equations of motion to solve a two-body pursuit problem.

Pause and Predict 3.4.1 |
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A ball is at rest at the top of an incline. When it is released, it rolls down the incline in the –x direction. If it takes 3 s to reach a speed of 6.5 m/s, what is the ball’s average acceleration? |

(a) -2.2 m/s^{2} |

(b) 2.2 m/s^{2} |

(c) -3.5 m/s^{2} |

(d) 3.5 m/s^{2} |

(e) 0 m/s |

Pause and Predict 3.4.2 |
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You drop a ball from a height of 35.0 m and it falls with a constant acceleration due to gravity of 9.80 m/s^{2}. What is the ball’s velocity after freely falling for 2.50 s? |

(a) 14.0 m/s |

(b) -14.0 m/s |

(c) 24.5 m/s |

(d) -24.5 m/s |

(e) 0 m/s |

### Kinematic Equations for Motion with Constant Acceleration

**Practice!**

Practice 3.4.1 |
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A car was parked on a steep hill and the driver forgot to set the emergency brake. As a result, the car rolled down the hill and crashed into a parked truck. If the car was moving at 10 mph (4.5 m/s) when it hit the truck 7.0 seconds after it began to move, what was the car’s average acceleration while it rolled down the hill? Define the positive x direction to be down the hill. |

(a) 1.4 m/s^{2} |

(b) -1.4 m/s^{2} |

(c) 0.6 m/s^{2} |

(d) -0.6 m/s^{2} |

(e) 1.6 m/s^{2} |

(f) -1.6 m/s^{2} |

Practice 3.4.2 |
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In a carnival game you roll a ball up a ramp and into a basket. During one attempt, you roll the ball so that it has a velocity of 6.0 m/s at the bottom of the ramp and 2.0 m/s at the top. If it takes 1.5 s for the ball to roll from the bottom to the top of the ramp, the ball’s average acceleration is… |

(a) -5.3 m/s^{2} |

(b) 5.3 m/s^{2} |

(c) -4.0 m/s^{2} |

(d) 4.0 m/s^{2} |

(e) -2.7 m/s^{2} |

(f) 2.7 m/s^{2} |

Practice 3.4.3 |
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Which of these equations is the correct mathematical representation of this position versus time graph? |

(a) |

(b) |

(c) |

(d) |

Practice 3.4.4 |
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Which of these equations is the correct mathematical representation of this velocity versus time graph? |

(a) |

(b) |

(c) |

(d) |

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

A car is traveling with an initial velocity *v*_{0} = 65 mph. At *t* = 0, the driver puts on the brakes, which slows the car at a rate of *a*_{b} = 6.5 m/s^{2}. At what time *t*_{f} does the car stop, and how much farther *x*_{f} does it travel?

You are traveling along an interstate highway at 37.0 m/s (about 83 mph) when a truck stops suddenly in front of you. You immediately apply your brakes and cut your speed in half after 4.0 s.

(a) What was your acceleration, assuming it was constant?

(b) How long did it take you to stop from the time you started to apply the brakes?

(c) What was your stopping distance?

At the instant a traffic light turns green, a car that has been waiting at the intersection starts ahead with a constant acceleration of 2.8 m/s^{2}. At the same instant a truck, traveling at a constant speed of 20.0 m/s, overtakes and passes the car.

(a) How far beyond its starting point does the car overtake and pass the truck?

(b) How fast is the car traveling when it overtakes the truck?