WORKSHEET:
Position and Velocity Functions

The purpose of this worksheet is to give you practice graphing position functions, and to practice the concept of velocity.

• For each story below, create a graph that matches the story.
• All the stories have the same set-up:
  you live on a long, straight, road (i.e., a number line).
  Your house is at position $0$.
  You always leave your house and turn right, which is the positive direction (i.e., towards $1, 2, ...$).
• Let $p(t)$ denote the position of the car at time $t$;
  put $p(t)$ along the vertical axis, and $t$ along the horizontal axis.
• Clearly label the times that are indicated in the stories ($a < b < c$ etc.) on your graphs.
  You may choose the locations of these times!
• When you are done with the the graphs, answer the questions that follow.

1. 1. Leave your house at $t = 0$. Gradually speed up so that you are going $50$ mph at $t = a$.
   2. Drive at a constant speed until $t = b$.
   3. Gradually slow down until you reach $30$ mph at $t = c$.
   4. Drive at a constant speed until $t = d$.
   5. Come quickly to a stop at $t = e$.
   6. Back up the car at a constant speed of $10$ mph until $t = f$.
   7. Go forward again until $t = g$.
2. ‘the Police Car’
   1. Leave your house at $t = 0$. Gradually speed up to $70$ mph at $t = a$.
   2. See a police car ahead of you on the road. Step on the brakes at $t = b$.
   3. Manage to get down to $50$ mph (the speed limit) by the time you pass the police car at $t = c$.
   4. After the police car is out of sight, gradually increase your speed so that you are going $70$ mph again at $t = d$.
   5. While you're thinking “Whoa. Close call!” you see another police car—too late! The flashing lights go on. The police car pulls you over, so you slow down quickly; at $t = e$ you pull over to the side of the road.
   6. You remain at the side of the road while the police person writes out a ticket. At $t = f$, you say good-bye and slowly pull back into traffic.
   7. You accelerate to $50$ mph for the rest of the trip, until you start slowing down at $t = g$ to get ready to stop.
   8. You are at a complete stop at $t = h$.
3. ‘I always forget things!’
   1. Leave your house at $t = 0$. You're really tired today and bummed out because of yesterday's speeding ticket, so you gradually speed up to $40$ mph at $t = a$.
   2. Drive at a constant speed with your thoughts wandering until $t = b$.
   3. At $t = c$, you suddenly realize that you've forgotten a homework assignment that needs to be passed in today. So, you quickly come to a complete stop by $t = d$.
   4. You turn around, and speed up to $50$ mph by $t = e$.
   5. You drive at $50$ mph until you start to slow down for your driveway at $t = f$.
   6. You slow down, coming to a complete stop at $t = g$.
   7. You run into your house and grab your homework. While there, the phone rings, which takes a few more minutes. Then, you get back out to your car at $t = h$ and take off again.
   8. You're now late, so you quickly speed up to $55$ mph by $t = i$.
   9. You drive at $55$ mph until you slow down and come to a complete stop at your final destination by $t = f$.
4. Please answer the following questions.
   In each case, include both WORDS and a SKETCH.
   1. What can you say about the slopes of the tangent lines when you're travelling forward (to the right, away from your house)?
   2. What can you say about the slopes of the tangent lines when you're travelling backwards (to the left, towards your house)?
   3. What can you say about the slopes of the tangent lines when you're travelling FAST to the right?
   4. What can you say about the slopes of the tangent lines when you're travelling SLOWLY to the right?
   5. What can you say about the slopes of the tangent lines when you're travelling FAST to the left?
   6. What can you say about the slopes of the tangent lines when you're travelling SLOWLY to the left?
   7. What can you say about the slopes of the tangent lines when you're not moving?
   8. What can you say about the slopes of the tangent lines when you're speeding up, going to the right? (Include a sketch of the shape of the curve.)
   9. What can you say about the slopes of the tangent lines when you're slowing down, going to the right? (Include a sketch of the shape of the curve.)
   10. What can you say about the slopes of the tangent lines when you're speeding up, going to the left? (Include a sketch of the shape of the curve.)
   11. What can you say about the slopes of the tangent lines when you're slowing down, going to the left? (Include a sketch of the shape of the curve.)