speed limit on freeway is 36 m/s and speed limit on service drive is 9 m/s. Mr. Scotts focus has a mass of 900 kg. Since friction depends on so many factors such as the condition of the vehicle and direction of wind, the civil engineer neglects it.
1. How deep should the freeway be so that a car that is going the speed limit at the top of the entrance ramp will be going the speed limit when it reaches the bottom (without touching gas pedal)
2. How deep should the freeway be so that a car going the speed limit at the bottom of the exit ramp will be going at the speed limit when it reaches the top?
3. Compare the results for the exit and entrance ramps. Which height is better in general?
4. the freeway will be used for both mini coopers and mack trucks. how will the different masses effect ones choice of an ideal depth for the freeway?
W=0.5M*V2^2-0.5M*V1^2 = 0.5M(V2^2-V1^2)=
450(9^2-36^2) = -546,750 J.
mg*h = -546,750.
8820h = 546,750.
h = 62. m.
2. W = 450(36^2-9^2) = -546,750 J.
Note: The negative sign means the system is doing the work.
To answer these questions, we need to consider the principles of mechanical energy conservation. When a car moves without using the gas pedal, it converts its potential energy to kinetic energy, while also losing some energy due to friction and air resistance.
1. To determine the depth for the freeway entrance ramp, we can equate the potential energy at the top of the ramp to the kinetic energy at the bottom, assuming no energy loss due to friction or air resistance. The formula for potential energy is given by mgh, where m is the mass of the car, g is the gravitational acceleration, and h is the height of the ramp. The kinetic energy of the car is given by (1/2)mv^2, where v is the speed limit on the freeway. So we can set up the equation: mgh = (1/2)mv^2. We can then solve for h by rearranging the equation as h = (1/2)v^2/g.
2. Similarly, to determine the depth for the freeway exit ramp, we can equate the kinetic energy at the bottom of the ramp to the potential energy at the top. The equation will be the same as in question 1, except we solve for h again.
3. By comparing the results for the entrance and exit ramps, we can determine which height is better in general. If the height for the entrance ramp is greater than the height for the exit ramp, then the car will gain more speed going down the entrance ramp and reach the speed limit without touching the gas pedal at the bottom. This would be a desirable design for efficient acceleration. However, if the height for the exit ramp is greater, the car will lose more speed going up the ramp, which could be desirable for controlling speed and ensuring safety.
4. The different masses of the vehicles, such as mini coopers and mack trucks, will have an impact on the choice of an ideal depth for the freeway. Heavier vehicles will have more potential energy at the top of the ramps, resulting in a greater need for depth to accommodate their higher kinetic energy at the bottom. Therefore, the ideal depth of the freeway should be chosen considering the range of vehicle masses expected to use it, to ensure a safe and efficient design for all types of vehicles.