You are getting ready to take a vacation at your cabin near a lake and will use

Question

You are getting ready to take a vacation at your cabin near a lake and will use your truck to tow your boat to the cabin. Your boat sits on a trailer and the trailer is attached to your truck using a short cable that runs at an angle of 30° to the horizontal. The boat and trailer together weigh 2000 lbs. and your truck weighs 4000 lbs. The cable can exert up to 500 lbs. before breaking. Because you are worried about the cable breaking while you are driving around town, you decide to calculate the minimum amount of time you should take to accelerate your truck from a stop up to the 30 mph (13.4 m/s) speed limit. You assume that rolling friction behaves like kinetic friction and that the coefficient of rolling friction between your tires and the road is 0.10.

Explanation / Answer

given,

weight of truck = 4000 lbs or 17792.8864 N

weight of trailer + boat = 2000 lbs or 8896.4432 N

angle = 30 degree

maximum tension on the cable = 500 lbs or 2224.1108 N

coefficient of friction = 0.1

equation of motion of truck

T * cos(theta) + k * (Mg + T * sin(theta)) = Ma

2224.1108 * cos(30) + 0.1 * (17792.8864 + 2224.1108 * sin(30)) = 1814.369 * a

a = 2.1035 m/s^2

by first equation of motion

v = u + at

13.4 = 0 + 2.1035 * t

t = 6.37 sec

minimum amount of time = 6.37 sec

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