An engineer is designing a new on-ramp for a highway. because of space limitations, the on-ramp can have a radius no larger than 40 meters. they are planning to use asphalt, which they know when wet has a coefficient of static friction of around 0.25. the smallest car that they anticipate will be taking the on-ramp is 1000 kg (a prius) and the largest is about 9000 kg (a mining truck). the on-ramp is not banked and we assume that the vehicle will travel at a constant speed.

a. what kind of acceleration will the car experience as it travels through the on-ramp?
b. calculate the maximum speed that each of the vehicles can take on the on-ramp. box your answers.
c. which of the two vehicles will have the easier time taking the on-ramp? (use the language of physics.)
d. what angular speed will the vehicles have as they travel on the on-ramp?
e. explain why banking the track makes it easier to drive through the on-ramp at higher speed. (use the language of physics.)

a) a = 2.45 m / s² , b)  v = 9.9 m / s , c) the vehicle that takes the curve the easiest is the light one , d) w = 9.8 rad / s

Explanation:

We can solve this problem with Newton's second law, with the ramp not inclined we will use a reference system with the y-axis and the x-axis in the radial direction

Y Axis  

         N-W = 0

         N = W

X axis

        fr = m a

The acceleration is centripetal

        a = v2 / r

The friction force is given by

       fr =  μ N

      μ mg = m a

      a = μ g

a) we calculate

       a = 0.25 9.8

       a = 2.45 m / s²

b) fr = m a

     μ mg = m v² / r

     v = √ (μ g r)

     v = √ (0.25 9.8 40)

     v = 9.9 m / s

c) the two vehicles take the ramp at the same speed, but the heavier vehicle requires more force to change its trajectory and therefore has more difficulty in the curve. Therefore the vehicle that takes the curve the easiest is the light one

d) angular velocity is related to linear velocity

        v = w r

       w = v / r

       w = 9.8 rad / s

e) by tilting the curve part of the normal is in the direction of the center of the curve, radial direction, therefore this component of the normal contributes and keeps the vehicle in the curve and the friction force can be less, so which can we take the curve faster if you slide

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