practice

1. Cars cruise down an expressway at 25 m/s. Engineers want to design an interchange for a deceleration of −2.0 m/s² over 3.0 s.
   1. What velocity will cars have at the end of the approach?
   2. What minimum approach length will satisfy these requirements?
   3. What maximum velocity could a car entering the interchange have and still be able to exit at the intended velocity? (Assume an extreme deceleration of four times the usual rate.)
2. A car with an initial velocity of 60 mph needs 144 feet to come to a complete stop. Determine the stopping distance of this same car, under identical conditions with an initial velocity of 30 mph, 20 mph, and 10 mph.
3. A typical commercial jet airliner needs to reach a speed of 180 knots before it can take off. (A knot is a nautical mile per hour and is very nearly equal to half a meter per second.) If such a plane spends 30 s on the runway estimate …
   1. its acceleration.
   2. the minimum runway length.
4. Write something completely different.

conceptual

1. On well-engineered highways obstacles like bridges, retaining walls, and guardrails are often protected by large yellow barrels filled with water or sand called impact attenuators. Should a wayward car find itself on on a collision course with a relatively immoveable obstacle, an impact attenuator would help reduce the severity of the crash. If one of these barrels provided sufficient protection for an impact at 30 km/h, how many barrels would provide sufficient protection for an impact at …
   1. 60 km/h?
   2. 90 km/h?
   3. 120 km/h?
2. In a series of tests conducted by the Insurance Institute for Highway Safety (IIHS), a passenger sedan traveling 55 mph came to a stop in 133 feet while a sleeper cab tractor with a loaded trailer required 196 feet. (Answer the following questions without considering the units.)
   1. How do the braking accelerations of these vehicles compare?
   2. How do the braking times of these vehicles compare?

numerical

1. Determine the minimum thickness of a fully inflated air bag if it must stop a passenger moving at 14 m/s (31 mph) with an acceleration of less than 60 g.
2. By how much should the front end of a car be designed to crumple such that a front end collision at 30 m/s (67 mph) would not result in an average acceleration greater than 30 g?
3. At main engine cutoff (MECO), the Space Shuttle is at an altitude of 113 km (70 miles), traveling 7600 m/s (17,000 mph) relative to the earth. This occurs 7 minutes 40 seconds into the mission. Determine the downrange distance of the shuttle from lift off to MECO.
4. A moving driver not anticipating an accident can apply the brakes fully in about 0.5 s. It takes something on the order of 5 s to stop a car on the freeway once the brakes are locked. Calculate the minimum emergency stopping distance needed at 30 m/s (65 mph).
5. Researchers at MIT are attempting to build robots that mimic the swimming abilities of fish. Two robotic fish have been constructed as of March 2000: Robotuna and Robot Pike. Tuna and Pike are exceptionally proficient swimmers. According to one researcher, "My project here is to build a swimming Pike. The characteristics that I hope to demonstrate are very quick turning and fast acceleration from a stop. In the wild, Pike accelerate at rates from 8-12 g during a start from a standstill to 6 m/s. While we may not be able to achieve these wild numbers, half or a quarter of this acceleration would still demonstrate that flapping foil propulsion is certainly capable of higher accelerations than a propeller."
   1. How long does it take a wild pike to reach its top speed?
   2. What distance would a wild pike cover in this time?
   3. How long does it take an adequately designed Robot Pike to reach its top speed?
   4. What distance would an adequately designed Robot Pike cover in this time?
6. The following diagram and paragraph compares the performance of a magnetically levitated train (the Transrapid) to a conventional high speed passenger train (the ICE).

   
   [magnify]

   The Transrapid is not only fast, but it can also accelerate quickly to high speeds. 300 km/h (185 mph) can be reached after a distance of only 5 km (3 miles). Modern high-speed trains require more than 28 km (18 miles) and at least four times as long to reach the same speed.

   How does the acceleration of the Transrapid compare to the acceleration of the ICE? (Note: Given the data in the text and diagram, there are three slightly different answers to this question.)
7. During a thunderstorm, a tree is blown over into a narrow road. A truck traveling at 16.0 m/s is 32.1 m away from the tree when the driver hits the brakes. After 4.0 s of braking the front bumper of the truck hits the tree. How serious was the collision with the fallen tree? Justify your answer with an appropriate calculation.
8. Two cars are adjacent to each other on a four lane highway. The first car accelerates uniformly from rest at 3.0 m/s² the moment the light changes to green. The second car approaches the intersection already moving at 18 m/s and is beside the first car at the instant the light changes. It then continues driving with a constant velocity.
   1. When are the two cars side by side again?
   2. When do they have the same velocity?
9. In soccer, the ball is placed 11 m from the goal line during a penalty kick. A good player can kick the ball at 29 m/s. Determine the acceleration required of the goalkeeper to block this ball given that the width of the goal is 7.32 m (8 yards).
10. A custom 1972 Vauxhall Victor modified by Andy Frost of Wolverhampton, UK is reported by some to be the world's fastest street legal car. That's not quite right. It doesn't have the highest top speed, but it probably does have the highest acceleration. Read this transcript of a report from the motoring magazine show Fifth Gear.

    Meet Andy Frost, a 45 year old automatic transmission specialist and creator of Red Victor 1. [Rapid cuts between shots of the car.] This has a 9.3 litre V-8. It's got 2,200 brake horsepower and does nought to sixty in one second. That's one second. Still not impressed? Watch this. [Telephoto shot of a quarter mile test.] The McLaren F1 can do the quarter mile in 11.1 seconds. This does it in 7.8 -- a record.

    Red Victor 1	Quarter Mile Time: 7.80 s

    Note:
    • John Lingenfelter reached a top speed of 254.76 mph (410.00 kph, 113.89 m/s) in 1989 driving a Callaway Corvette Sledgehammer. This is probably the fastest street legal car that ever existed.
    • The McLaren F1 is often used as the standard for high performance cars, but the Bugatti Veyron 16.4 outperforms the McLaren on most tests. The Bugatti has a reported nought to sixty time of 2.5 seconds. Red Victor 1 beats this easily, so it is probably the car with the greatest acceleration.
    Given all of this information, determine the following quantities for Red Victor 1.

    Determine the average speed during the quarter mile run in … mph km/h m/s Determine the final speed during the quarter mile run (assuming uniform acceleration) in … mph km/h m/s How do these numbers compare to the top speed of the Callaway Corvette Sledgehammer?

    Determine the average acceleration during the nought to sixty test in … m/s g's Determine the average acceleration during the quarter mile run in … m/s g's Why do these two calculations yield noticeably different values and what does this say about the results of your final speed calculation in part b?

calculus

1. An object's position is described by the following polynomial for 10 s.

   s = t3 − 15t2 + 54t

   Where s is in meters and t is in seconds. Determine …
   1. the object's velocity as a function of time
   2. the object's acceleration as a function of time
   3. the object's maximum velocity
   4. the object's minimum velocity
   5. the time when the object was moving backward
   6. the time(s) when the object was at the origin (s = 0 m)
   7. the time(s) when the object returned to its starting position
   8. the object's average velocity
   9. the object's average speed
2. An object's velocity, v, in meters per second is described by the following function of time, t, in seconds for a substantial length of time …

   v = 4t (4 − t) + 8

   Assuming the object is located at the origin (s = 0 m) when t = 0 s determine …
   1. the object's position, s, as a function of time
   2. the object's acceleration, a, as a function of time
   3. the object's maximum velocity
   4. if and when when the object stops
   5. if and when the object returns to the origin (s = 0 m)
3. The following equations state displacement as a function of time. Derive the subsequent equations for velocity and acceleration as functions of time. (The symbols A, f, j, k, x₀, π and τ are all constants.)
   1. x = ⅙jt³
   2. x = A sin(2πft)
   3. x = x₀e^−t/τ
4. A crude mathematical model of tunneling is represented by the equation …

   v =	k
   	x

   where v is the tunneling speed; x is the length of the tunnel; and k is a constant.
   1. In what way (increase or decrease) does the tunneling speed change as the tunnel gets longer? What engineering aspect of tunneling is causing this change?
   2. Determine the following quantities as a function of time …
      1. tunnel length
      2. tunneling speed
      3. tunneling acceleration
      (This is a somewhat difficult problem for students who have just started learning calculus.)
5. A simplified model of a car accelerating from rest along a straight path is given by the following equation …

   v(t) = A(1 − e^−bt)

   Where v(t) is the instantaneous speed of the car in feet per second, t is the time in seconds, and A and b are constants.
   1. speed
      1. What are the units in the coefficients A and b?
      2. What is the physical meaning of the coefficent A?
      3. What is the speed of the car at t = 0 s?
      4. What is the asymptote of this function as t → ∞?
      5. Sketch a graph of speed vs. time. Include the value of v(0) and the asymptote of v as t → ∞.
   2. position
      1. Derive an equation s(t) for the instantaneous position of the car as a function of time. (Be sure that your function has the value s = 0 when t = 0.)
      2. What is the asymptote of this function as t → ∞?
      3. What is the physical meaning of the slope of this asymptote?
      4. Sketch a graph of position vs. time. Include the value of s(0) and the asymptote of s as t → ∞.
   3. acceleration
      1. Derive an equation a(t) for the instantaneous acceleration of the car as a function of time.
      2. What is the acceleration of the car at t = 0 s?
      3. What is the asymptote of this function as t → ∞?
      4. Sketch a graph of acceleration vs. time. Include the value of a(0) and the asymptote of a as t → ∞.
   4. numbers
      Apply this model to a real but exceptional car -- the Red Victor 1. This car has a zero-to-sixty time of about one second and a quarter mile time of about eight seconds. In other words, let …

      v(1 sec) = 88 ft/sec	and	s(8 sec) = 1320 ft

      then determine …
      1. the values of the coefficients A and b [I think this can only be done using a fancy calculator.]
      2. the maximum speed, and
      3. the maximum acceleration.
6. A crude mathematical model of tunneling is represented by the equation …

   v =	k
   	x

   where v is the tunneling speed; x is the length of the tunnel; and k is a constant.
   1. In what way (increase or decrease) does the tunneling speed change as the tunnel gets longer? What engineering aspect of tunneling is causing this change?
   2. Determine the following quantities as a function of time …
      1. tunnel length
      2. tunneling speed
      3. tunneling acceleration
      (This is a somewhat difficult problem for students who have just started learning calculus.)

statistical

1. braking-distance.txt
   The accompanying text file contains the stopping distances in feet for 123 different automobiles. The initial speed was 60 mph in the first column and 80 mph in the second column.
   1. Determine the equation of the line of best fit for this data set.
   2. Why does the slope of this line have the value that it does?
   3. Which proportionality in this section is relevant and what value does it predict for the slope?

investigative

1. Obtain the performance specifications for an airplane of any sort. Identify the relevant data and then calculate the tightest runway acceleration that this airplane could have and yet still take off safely. You might also need runway data for an airport capable of handling such a plane. Since there are several factors affecting take off, you should perform this calculation a few times for a range of different conditions. As an option, you could also try repeating this problem for landing instead of takeoff. That is, calculate the tightest runway deceleration that this airplane could have and yet still land safely.