## 2.16 Velocity and Acceleration

### Summary

Derivatives are used to analyze the motion of an object on a straight line. It is remarkable how starightforward it is to obtain a detailed description of the object's motion from considering properties of the first and second derivative.

By the end of your studying, you should know:

• The equation of motion for an object dropped (or thrown up and allowed to fall) near the surface of the earth.
• How to determine such characteristics of the motion of a falling object as when it hits the ground, or how high it goes if it was initially thrown up into the air.
• How to answer questions about an object traveling on the x-axis according to a given equation. Questions such as: When is the object at rest? When is it moving to the right? When is it accelerating? When is it speeding up? slowing down?

On-screen applet instructions: The position of a pickup truck traveling on the x-axis is shown at time t. Use the slider to change t, and observe the relationship between the position curve x(t), velocity curve v(t), and the direction of travel of the truck.

### Examples

A golfer on the moon (where gravitational acceleration equals 1.67 m/sec2) hits a ball whose initial velocity in the vertical direction is 30 meters per second. What is the maximum height the ball reaches?

A meteoroid falling to Earth is discovered when it is at an altitude of 9000 kilometers, traveling at a velocity of 70 kilometers per second. Assuming acceleration due to Earth's gravity is constant, and neglecting air resistance, how fast will the meteoroid be falling when it hits the ground? What will its acceleration be?

You drop a rock off Quechee Gorge Bridge and it hits the water below about 3.2 seconds later. Approximately how high is the bridge?

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.16 Velocity and Acceleration (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### Interesting Application

What will happen if these two cannons fire at each other at the same time?

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel