1) In this problem we analyze the phenomenon of "tailgating" in a car on a highway at high speeds. This means traveling too close behind the car ahead of you. Tailgating leads to multiple car crashes when one of the cars in a line suddenly slows down. The question we want to answer is: "How close is too close?"

To do this, let's suppose you are driving on the highway at a speed of 100 kilometers an hour (a bit more than 60 mph). The car ahead of you suddenly puts on its brakes. We need to calculate a number of things: how long it takes you to respond; how far you travel in that time; how far the other car traveled in that time.

First let's estimate how long it takes you to respond. Two times are involved: how long it takes from the time you notice something happening till you start to move to the brake, and how long it takes to move your foot to the brake. You will need a piece of paper (8.5x11), a meter stick, and a coin to do this.

Take a piece of paper and have a friend hold the paper from the middle of one of the short sides hanging straight down. Place your thumb and forefinger opposite the bottom of the paper. Have your friend release the paper suddenly and try to catch it with your thumb and forefinger. Measure how far the paper fell before you caught it. Do this three times and take the average distance. Assuming the paper was falling freely without air resistance (not bad for the paper falling sideways), calculate how much time it took before you caught it, t₁.

(a) Estimate the time it takes you to move your foot from the gas pedal to the brake to be t₂ = 0.5 s. Your reaction time is t₁ + t₂.
(b) If you brake hard and fast, you can bring a typical car to rest from 100 kph (about 60 mph) in 5 seconds. Calculate your acceleration, a₀, assuming that it is constant.
(c) Suppose the car ahead of you begins to brake with a (negative) acceleration a₀. How far will he travel before he comes to a stop? (Hint: How much time will it take him to stop? What will be his average velocity over this time interval?)
(d) You see him start to slow immediately (an unreasonable but simplifying assumption). If you are also traveling 100 kph, how far (in meters) do you travel before you begin to brake? If you can also produce the acceleration a₀ when you brake, what will be the total distance you travel before you come to a stop?
(e) If you don't notice the car ahead of you beginning to brake for 1 second, how much additional distance will you travel?
(f) Discuss, on the basis of these calculations, what you think a safe distance is to stay behind a car at 60 mph. Would you include a safety factor beyond what you have calculated here? How much?

2) An unrestrained child is playing on the front seat of a car traveling in a residential neighborhood at 35 km/h. (How many mph is this?) A small dog runs across the road and the driver applies the brakes, stopping the car and missing the dog. With what speed does the child strike the dashboard, presuming that the car stops before the child does so? Compare this speed with that of the world-record 100 m dash, which is run in about 10 s.

3) A small ball is projected into a narrow, frictionless tube bent into the arc of a circle and lying flat on a horizontal table at the point A as shown in the picture on the right. If you ignore the friction between the ball and the table, what path will the ball follow when it leaves the tube at B? Explain your reasoning.

4) The figure below shows the velocity graph of a cart moving on an air track. The track has a spring at one end and has its other end raised. The cart is started sliding up the track by pressing it against the spring and releasing it. The clock is started just as the cart leaves the spring. Take the direction the cart is moving in initially to be the positive x direction and take the bottom of the spring to be the origin.

Letters point to six points on the velocity curve. For the physical situations described below, identify which of the letters corresponds to the situation described. You may use each letter more than once, more than one letter may be used for each answer, or none may be appropriate. If none is appropriate, put the letter N.

(a) This point occurs when the cart is at its highest point on the track.
(b) At this point, the cart is instantaneously not moving.
(c) This is a point when the cart is in contact with the spring.
(d) At this point, the cart is moving down the track toward the origin.
(e) At this point, the cart has acceleration of zero.

5) The figure below represents the position vs. clock reading of the motion of two balls, A and B, moving on parallel tracks.

Carefully sketch the figure on your paper and answer the following questions:

(a) Mark with the symbol t_a along the t-axis any instant or instants at which one ball is passing the other.
(b) Which ball is moving faster at clock reading t_b?
(c) Mark with the symbol t_c along the t-axis any instant or instants at which the balls have the same velocity.
(d) Over the period of time shown in the diagram, which of the following is true of ball B? Explain your answer.

• (1) It is speeding up all the time.
• (2) It is slowing down all the time.
• (3) It is speeding up part of the time and slowing down part of the time.

6) In the figure below is shown a graph of the velocity of a young boy riding his bicycle as a function of time. Write a "story" describing the boy's actions that lead to this graph (keep it short!) and pose an end-of-chapter physics problem that could be solved using the graph.

7) A motorized dune buggy is moving along an elliptical track. A picture of the track with a choice of coordinate system is shown in the figure at the top of the next page. Each tic mark along the x and y axes represents a distance of 100 meters.

(a) The driver drives around the track three times keeping her speedometer reading at a constant 35 miles per hour. Redraw the ellipse and its coordinate system. During her second circuit of the ellipse, draw onto your picture vectors indicating her position (use a single arrow), her velocity (use a double arrow) and her acceleration (use a fat arrow) when she is at the positions

(b) As she begins her fourth circuit round the track, she begins to speed up when she reaches the point {600 m, 0} and continues speeding up at a uniform rate until she reaches the point {-600 m, 0} and is traveling at a speed of 50 mph. Draw another ellipse and draw velocity vectors (use a double arrow) to indicate her velocity when she is a few meters before the point {0, 400 m} and a few meters after that point. Draw her acceleration arrow (use a fat arrow) when she is at the position {0, 400 m}.
(c) A viewer proposes that the equations describing her motion during the first three circuits would be

with a = 600 m and b = 400 m. Do you agree with the viewer? If so, explain why you think so. If not, explain how you would correct the equation.

8) A cart can move to the right or left along a horizontal track (the positive part of the x axis) as shown in the figure below. Assume that friction is small enough that it can be ignored. A sonic range is used (as shown) to measure the position, velocity, and acceleration of the cart. The track is not necessarily flat or horizontal. In addition, the track may be tipped or the cart may be pulled or pushed.

(a) For the first run, the sonic ranger displays a graph of the velocity that looks like the graph labeled A shown at the right. Copy the graph and draw graphs showing what the sonic ranger would display for (i) the cart's position (ii) the cart's acceleration. In addition (iii) describe the motion of the cart in words.
(b) For the second run, the sonic ranger displays of graph of the velocity that looks like the graph labeled B shown at the right. Repeat the tasks of part (a) for this case.