Resistance, Tractive Effort, and Acceleration (Sections 2.2-2.7)
2.1 A new sports car has a drag coefficient of 0.30 and a frontal area of 21 ft2, and is traveling at 110 mi/h.
How much power is required to overcome aerodynamic drag if ȡ = 0.002378 slugs/ft3?
2.2 For Example 2.3, how far back from the front axle would the center of gravity have to be to ensure that the maximum tractive effort developed for front- and rear- wheel–drive options is equal (assume that all other variables are unchanged)?
2.3 A vehicle manufacturer is considering an engine
for a new sedan (CD = 0.34, Af = 22 ft2). The car is being designed to achieve a top speed of 100 mi/h on a paved surface at sea level (ȡ = 0.002378 slugs/ft3). The car currently weighs 2500 lb, but the designers initially
selected an underpowered engine because they did not account for aerodynamic and rolling resistances. If 2 lb of additional vehicle weight is added for each unit of horsepower needed to overcome the neglected resistance, what will be the final weight of the car if it is to achieve the 100-mi/h top speed?
2.4 A 2650-lb car is traveling at sea level at a constant speed. Its engine is running at 4500 rev/min and is producing 175 ft-lb of torque. It has a drivetrain efficiency of 90%, a drive axle slippage of 2%, 15- inch–radius wheels, and an overall gear reduction ratio of 3 to 1. If the car’s frontal area is 21.5 ft2, what is its drag coefficient?
2.5 A 3000-lb car has a maximum speed (at sea level
and on a level, paved surface) of 140 mi/h with 16-inch–radius wheels, a gear reduction of 3.5 to 1, and a drivetrain efficiency of 92%. It is known that at the
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car’s top speed the engine is producing 220 ft-lb of torque. If the car’s frontal area is 25 ft2, what is its drag coefficient?
2.6 A 3200-lb car (CD = 0.35, Af = 25 ft2, and ρ = 0.002378 slugs/ft3) has 14-inch–radius wheels, a drivetrain efficiency of 93%, an overall gear reduction ratio of 3.2 to 1, and drive axle slippage of 3.5%. The engine develops a maximum torque of 210 ft-lb at 3600 rev/min. What is the maximum grade this vehicle could ascend, on a paved surface, while the engine is developing maximum torque? (Assume that the available tractive effort is the engine-generated tractive effort.)
2.7 A 3400-lb car is traveling in third gear (overall gear reduction ratio of 2.5 to 1) on a level road at its top speed of 130 mi/h. The air density is 0.00206 slugs/ft3. The car has a frontal area of 19.8 ft2, a drag coefficient of 0.28, a wheel radius of 12.6 inches, a drive axle slippage of 3%, and a drivetrain efficiency of 88%. At this vehicle speed, what torque is the engine producing and what is the engine speed (in revolutions per minute)?
2.8 A rear-wheel–drive car weighs 2600 lb and has an
84-inch wheelbase, a center of gravity 20 inches above the roadway surface and 30 inches behind the front axle, a drivetrain efficiency of 85%, 14-inch–radius wheels, and an overall gear reduction of 7 to 1.
The car’s torque/engine speed curve is given by Me=6ne−0.045ne2
If the car is on a paved, level roadway surface with a coefficient of adhesion of 0.75, determine its maximum acceleration from rest.
2.9 Consider the car in Problem 2.8. If it is known that the car achieves maximum speed at an overall gear reduction ratio of 2.7 to 1 with a drive axle slippage of 3.5%, how fast would the car be going if it could achieve its maximum speed when its engine is producing maximum power?
2.10 An engineer designs a rear-wheel–drive car
(without an engine) that weighs 2000 lb and has a 100- inch wheelbase, drivetrain efficiency of 80%, 14-inch–
radius wheels, an overall gear reduction ratio of 10 to 1, and a center of gravity (without engine) that is 22 inches above the roadway surface and 55 inches behind the front axle. An engine that weighs 3 lb for each ft-lb of developed torque is to be placed in the front portion of the car. Calculations show that for every 20 lb of engine weight added, the car’s center of gravity moves
1 inch closer to the front axle (but stays at the same height above the roadway surface). If the car is starting from rest on a level paved roadway with a coefficient of adhesion of 0.8, select an engine size (weight and associated torque) that will result in the highest possible available tractive effort.
2.11 A 3000-lb car is traveling on a paved road with CD
= 0.35, Af = 21 ft2, and ρ = 0.002378 slugs/ft3. Its engine is running at 3000 rev/min and is producing 250 ft-lb of torque. The car’s gear reduction ratio is 4.5 to 1, drivetrain efficiency is 90%, drive axle slippage is 3.5%, and the wheel radius is 16 inches. What will the car’s maximum acceleration rate be under these conditions on a level road? (Assume that the available tractive effort is the engine-generated tractive effort.)
2.12 A rear-wheel–drive car weighs 3600 lb, has 15-
inch–radius wheels, a drivetrain efficiency of 95%, and an engine that develops 520 ft-lb of torque. Its wheelbase is 8.2 ft, and the center of gravity is 18 inches above the road surface and 3.3 ft behind the front axle. What is the lowest gear reduction ratio that would allow this car to achieve the highest possible acceleration from rest on good, dry pavement?
2.13 A newly designed car has a 9.0-ft wheelbase, is
rear-wheel drive, and has a center of gravity 18 inches above the road and 4.3 ft behind the front axle. The car weighs 2450 lb, the mechanical efficiency of the drivetrain is 90%, and the wheel radius is 14 inches. The base engine develops 200 ft-lb of torque, and a modified version of the engine develops 240 ft-lb of torque. If the overall gear reduction ratio is 8 to 1, what is the maximum acceleration from rest for the car with the base engine and for the car with the modified engine? (It is on good, dry, and level pavement.)
2.14 A rear-wheel–drive 3000-lb drag race car has a
200-inch wheelbase and a center of gravity 20 inches above the pavement and 140 inches behind the front axle. The owners wish to achieve an initial acceleration from rest of 22 ft/s2 on a level paved surface. What is the minimum coefficient of road adhesion needed to achieve this acceleration? (Assume γm = 1.00.)
2.15 If the race car in Problem 2.14 has a center of
gravity 32 inches above the roadway and is run on a pavement with a coefficient of adhesion of 1.0, how far back from the front axle would the center of gravity have to be to develop a maximum acceleration from rest of 1.0 g (32.2 ft/s2)? (Assume γm = 1.00.)
2.16 Consider the situation described in Example 2.5. If the vehicle is redesigned with wheels that have a 13- inch radius (assume that the mass factor is unchanged)
and a center of gravity located at the same height but at the midpoint of the wheelbase, determine the acceleration for front- and rear-wheel–drive options.
Braking and Stopping Distance (Section 2.9)
2.17 If the car in Example 2.9 had CD = 0.45 and Af = 25 ft2, what is the difference in minimum theoretical stopping distances with and without aerodynamic resistance considered (all other factors the same as in Example 2.9)?
2.18 A 3500-lb vehicle (CD = 0.38, Af = 26 ft2, ρ = 0.002378 slugs/ft3) is driven on a surface with a coefficient of adhesion of 0.5, and the coefficient of rolling friction is approximated as 0.015 for all speeds.
Assuming minimum theoretical stopping distances, if the vehicle comes to a stop 260 ft after brake application on a level surface and has a braking efficiency of 0.82, what was its initial speed (a) if aerodynamic resistance is considered and (b) if aerodynamic resistance is ignored?
2.19 A level test track has a coefficient of road
adhesion of 0.80, and a car being tested has a coefficient of rolling friction that is approximated as 0.018 for all speeds. The vehicle is tested unloaded and achieves the theoretical minimum stop in 180 ft (from brake application). The initial speed was 60 mi/h. Ignoring aerodynamic resistance, what is the unloaded braking efficiency?
2.20 A driver is traveling at 90 mi/h down a 3% grade on good, wet pavement. An accident investigation team noted that braking skid marks started 410 ft before a parked car was hit at an estimated 45 mi/h. Ignoring air resistance, and using theoretical stopping distance, what was the braking efficiency of the car?
2.21 A small truck is to be driven down a 4% grade at 70 mi/h. The coefficient of road adhesion is 0.95, and it is known that the braking efficiency is 80% when the truck is empty and decreases by one percentage point for every 100 lb of cargo added. Ignoring aerodynamic resistance, if the driver wants the truck to be able to achieve a minimum theoretical stopping distance of 275 ft from the point of brake application, what is the maximum amount of cargo (in pounds) that can be carried?
2.22 Consider the conditions in Example 2.11. The car has W = 3500 lb, CD = 0.5, Af = 25 ft2, ρ = 0.002378 slugs/ft3, and a coefficient of rolling friction approximated as 0.018 for all speed conditions. If aerodynamic resistance is considered in stopping, estimate how fast the car will be going when it strikes
the object on a level and a +5% grade [all other conditions (speed, etc.) as described in Example 2.11].
2.23 A race car with a 106-inch wheelbase has its
weight evenly distributed between front and rear axles.
At 150 mi/h, on a race track with ȝ = 1.0, the optimal brake force has 67.32% of the braking force on the front brakes. A new racing tire generates ȝ = 1.2. At 150 mi/h, what percentage of the braking force should now be allocated to the front to achieve optimal braking?
2.24 A car is traveling up a 2% grade at 70 mi/h on
good, wet pavement. The driver brakes to try to avoid hitting stopped traffic that is 250 ft ahead. The driver’s reaction time is 0.5 s. At first, when the driver applies the brakes, a software flaw causes the anti-lock braking system to fail (brakes work in non-anti-lock mode with 80% efficiency), leaving 80 ft skid marks. After the 80 ft skid, the anti-lock brakes work with 100% efficiency.
How fast will the driver be going when the stopped traffic is hit if the coefficient of rolling resistance is constant at 0.013? (assume minimum theoretical stopping distance and ignore aerodynamic resistance) 2.25 A car is traveling at 76 mi/h down a 3% grade on poor, wet pavement. The car’s braking efficiency is 90%. The brakes were applied 320 ft before impacting an object. The car had an antilock braking system, but the system failed 200 ft after the brakes had been applied (wheels locked). What speed was the car traveling at just before it impacted the object? (Assume theoretical stopping distance, ignore air resistance, and let frl = 0.015.)
2.26 A driver traveling down a 4% grade collides with a roadside object in rainy conditions, and is issued a ticket for driving too fast for conditions. The posted speed limit is 65 mi/h. The accident investigation team determined the following: The vehicle was traveling 40 mi/h when it struck the object, braking skid marks started 205 ft before the struck object, the pavement is in good condition, and the braking efficiency of the vehicle was 93%. Using theoretical stopping distance, assuming aerodynamic resistance is negligible, and with the coefficient rolling resistance approximated as 0.015, should the driver appeal the ticket? Why or why not?
2.27 A driver is traveling 68 mi/h on a road with a
−3% grade. There is a stalled car on the road 1000 ft ahead of the driver. The driver’s vehicle has a braking efficiency of 90%, and it has antilock brakes. The road is in good condition and is initially dry, but it becomes wet 160 ft before the stalled car (and stays wet until the
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car is reached). What is the minimum distance from the stalled car at which the driver could apply the brakes and still stop before hitting it? (Assume theoretical stopping distance, ignore air resistance, and let frl = 0.013.)
2.28 A car is traveling at 70 mi/h on a level section of road with good, wet pavement. Its antilock braking system (ABS) only starts to work after the brakes have been locked for 100 ft. If the driver holds the brake pedal down completely, immediately locking the wheels, and keeps the pedal down during the entire process, how many feet will it take the car to stop from the point of initial brake application? (The braking efficiency is 80% with the ABS not working and 100%
with the ABS working. Use theoretical stopping distance and ignore air resistance. Let frl = 0.02 when the brakes are locked, but compute the frl once the ABS becomes active.)
2.29 Two cars are traveling on level terrain at 60 mi/h on a road with a coefficient of adhesion of 0.8. The driver of car 1 has a 2.5-s perception/reaction time and the driver of car 2 has a 2.1-s perception/reaction time.
Both cars are traveling side by side and the drivers are able to stop their respective cars in the same distance after first seeing a roadway obstacle (perception and reaction plus vehicle stopping distance). If the braking efficiency of car 2 is 0.78, determine the braking efficiency of car 1. (Assume minimum theoretical stopping distance and ignore aerodynamic resistance.)
2.30 An engineering student is driving on a level
roadway and sees a construction sign 500 ft ahead in the middle of the roadway. The student strikes the sign at a speed of 25 mi/h. If the student was traveling at 55 mi/h when the sign was first spotted, what was the student’s associated perception/reaction time (use practical stopping distance)?
2.31 An engineering student claims that a country road can be safely negotiated at 65 mi/h in rainy weather.
Because of the winding nature of the road, one stretch of level pavement has a sight distance of only 510 ft.
Assuming practical stopping distance, comment on the student’s claim.
2.32 A driver is traveling at 52 mi/h on a wet road. An object is spotted on the road 415 ft ahead and the driver is able to come to a stop just before hitting the object.
Assuming standard perception/reaction time and practical stopping distance, determine the grade of the road.
2.33 A test of a driver’s perception/reaction time is
being conducted on a special testing track with wet
pavement and a driving speed of 50 mi/h. When the driver is sober, a stop can be made just in time to avoid hitting an object that is first visible 385 ft ahead. After a few drinks under exactly the same conditions, the driver fails to stop in time and strikes the object at a speed of 30 mi/h. Determine the driver’s perception/reaction time before and after drinking. (Assume practical stopping distance.)
Acceleration and Braking (Sections 2.7 and 2.9) 2.34 On a level test track, a car with anti-lock brakes and 90% braking efficiency is determined to have a theoretical stopping distance (ignoring aerodynamic resistance) of 408 ft (after the brakes are applied) from 100 mi/h. The car is rear-wheel drive with a 110 inch wheel base, weighs 3200 lb, and has a 50/50 weight distribution (front to back), a center of gravity that is 22 inches above the road surface, an engine that generates 300 ft-lb of torque, an overall gear reduction of 8.5 to 1 (in first gear), a wheel radius of 15 inches and a driveline efficiency of 95%. What is the maximum acceleration from rest of this car on this test track?
Multiple Choice Problems (Multiple Sections) 2.35 A 2500-lb vehicle has a drag coefficient of 0.35 and a frontal area of 20 ft2. What is the minimum tractive effort required for this vehicle to maintain a 70 mi/h speed on a 5% upgrade through an air density of 0.002045-slugs/ft3?
a) 217.9 lb b) 172.0 lb c) 136.9 lb d) 135.1 lb
2.36 A car is traveling at 20 mi/h on good, dry
pavement at 5000 ft elevation. The front-wheel-drive car has a drag coefficient of 0.30, a frontal area of 20 ft2 and a weight of 2500 lb. The wheelbase is 110 inches and the center of gravity is 20 inches from the ground, 50 inches behind the front axle. The engine is producing 95 ft-lb of torque and is in a gear that gives an overall gear reduction ratio of 4.5. The radius of the drive wheels is 14 inches and the mechanical efficiency of the drivetrain is 90%. What would the acceleration of the car be if the driver was accelerating quickly to avoid a collision?
a) 3.65 ft/s2 b) 15.53 ft/s2 c) 15.90 ft/s2 d) 3.48 ft/s2
2.37 A car is traveling at 60 mi/h on good, wet pavement. It has a wheelbase of 110 inches with the center of gravity 50 inches behind the front axle and at a height of 24 inches above the pavement surface.
Determine the percentage of braking force that the braking system should allocate to the rear axle.
a) 74.5%
b) 65.4%
c) 25.5%
d) 34.6%
2.38 A truck traveling at 75 mi/h has a braking
efficiency of 70%. The coefficient of road adhesion is 0.80. Ignoring aerodynamic resistance, determine the theoretical stopping distance on a level grade.
a) 340.9 ft b) 180.6 ft c) 425.6 ft d) 338.6 ft
2.39 A child accidentally runs into the street in front of an approaching vehicle. The vehicle is traveling at 40 mi/h. Assuming the road is level, at what distance must the driver first see the child to stop just in time?
a) 153.7 ft b) 300.3 ft c) 318.8 ft d) 146.7 ft
2.40 A car is traveling at sea level at 78 mi/h on a 4%
upgrade before the driver sees a fallen tree in the roadway 150 feet away. The coefficient of road adhesion is 0.8. The car weighs 2700 lb, has a drag coefficient of 0.35, a frontal area of 18 ft2, and a coefficient of rolling friction approximated as 0.017 for all speed conditions. The car has an antilock braking system that gives it a braking efficiency of 100%. If the driver first applies the brakes 150 ft from the tree, how fast will the car be traveling when it reaches the tree?
Include the effect of aerodynamic resistance.
a) 49.5 mi/h b) 48.8 mi/h c) 50.5 mi/h d) 47.7 mi/h