The Physics
Hypertextbook
Opus in profectus

# Equations of Motion

## Problems

### 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 that lasts 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 with an initial velocity of…
1. 30 mph
2. 20 mph
3. 10 mph
Note: The rate of change of velocity is not affected by inital velocity in this problem. Fast cars and slow cars slow down at the same rate.
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.

### 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 130 feet while a sleeper cab tractor with a loaded trailer required 195 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 (67 mph).
5. Pike are long-bodied predatory freshwater fish that are exceptionally proficient swimmers. Robot Pike is an autonomous undersea vehicle (AUV) that tries to reproduce some of a real pike's abilities — and hopefully excede the abilities of more conventional AUVs. According to one of the robot's designers…

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.

AUVAC, date unknown

1. What minimum time does a wild pike need to reach its top speed?
2. What minimum distance does a wild pike need to reach its top speed?
3. What maximum time could a Robot Pike take to reach its top speed and still be considered adequately designed?
4. What maximum distance could a Robot Pike take to reach its top speed and still be considered adequately designed?
6. The following paragraph compares the performance of a magnetically levitated train (the Transrapid) to a conventional high speed passenger train (the ICE).

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.

ThyssenKrupp Transrapid GmbH, 2002

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 acceptable 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 was reported by some to be the world's fastest street legal car from 2006 to 2014. This statement is not entirely true. It was only the fastest car in the quarter mile. There are several street legal cars that can go faster if given enough room. Mr. Frost's car is more impressive for its acceleration. It probably held the world record in that category for eight years.

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.

Fifth Gear, 2006

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 one of the fastest street legal cars that ever existed.
• The Bugatti Veyron 16.4 has a reported nought to sixty time of 2.5 seconds. Andy Frost's car beats this easily, so Red Victor 1 is probably the car with the greatest acceleration.
Given all of this information, determine the following quantities for Red Victor 1.
1. Determine the average speed during the quarter mile run in…
1. mph
2. km/h
3. m/s
2. Determine the final speed during the quarter mile run (assuming uniform acceleration) in…
1. mph
2. km/h
3. m/s
3. How do the values calculated in parts a. and b. compare to the top speed of the Callaway Corvette Sledgehammer?
4. Determine the average acceleration during the nought to sixty test in…
1. m/s
2. g's
5. Determine the average acceleration during the quarter mile run in…
1. m/s
2. g's
6. 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?
11. A 747 commercial jet airliner traveling at 70 m/s touches down on a runway. The jet slows to rest at a rate of –2.0 m/s2.
1. Calculate the average velocity of the 747 as it is brought to rest on the runway.
2. Calculate the total distance the 747 travels on the runway as it is brought to rest.
3. Calculate the time required to stop the 747 on the runway.
4. An F18 fighter jet traveling at a speed of 140 m/s touches down 180 seconds later on the same runway as the 747. What acceleration would the F18 need to come to a complete stop in the same amount of space as the 747?

### algebraic

1. Prove that the difference of two adjacent squares is always an odd number (for example 9 − 4 = 5 or 16 − 9 = 7). What relation could this possibly have to one-dimensional motion with constant acceleration? (Galileo was probably the first person to make this connection.)

### statistical

1. braking-distance.txt
The accompanying text file contains the stopping distances in meters for 123 different automobiles. The initial speed was 60 mph (26.82 m/s) in the first column and 80 mph (35.72 m/s) in the second column. Make a distance-distance graph and analyze it.
1. Make a scatterplot of stopping distance at 80 mph vs. stopping distance at 60 mph.
2. Determine the line of best fit for this graph.
3. Discuss the significance of the slope of this best fit line. That is, how are braking distances affected by initial speed? (This question is somewhat difficult.)
Make an acceleration-acceleration graph and analyze it.
1. Compute the average braking acceleration for each car…
1. starting at 60 mph
2. starting at 80 mph
2. Make a scatterplot of braking acceleration from 80 mph vs braking acceleration from 60 mph.
3. Determine the line of best fit for this graph.
4. Discuss the significance of the slope of this best fit line. That is, how are braking accelerations affected by initial speed? (This question is not so difficult.)