## Problems

A length of road is straight, and it has a profile in the x-y plane described by where 0= x=3 miles=15,840 ft; f(x) and x are in feet.

(a) Plot the road profile in the x-y plane for 0=x=15,840 ft.

(b) Derive an expression for B(x). Calculate B(500 ft).

(c) Derive an expression for percent grade (x). Calculate percent grade (500 ft).

(d) Derive an expression for tangential road length s(x), such that s(0)=0. Calculate s(500 ft).

(e) Can you find an expression for x(s), i.e., can you express x as a function of s? Show some steps in your attempt.

A straight roadway has a profile in the x-y plane given by

f(xf) and xf are measured in feet.

(a) Derive an expression for B(xf). Calculate B(1 mi).

(b) Calculate the tangential road length, s from 0 to 2 mi.

An electric vehicle has the following parameter values:

a, = w?. .vj. r, = ij..\.-i... ^ I|;-. c,- D.d-::1.'. c = i o in- -Vm-

(a) EVat rest—The EV is stopped at a stop sign at a point in the road where the grade is +15%. The tractive force of the vehicle is supplied by the vehicle brakes.

(i) Calculate the tractive force necessary for zero rolling resistance. (The vehicle is at rest.)

(ii) Calculate the minimum tractive force required from the brakes to keep the EV from rolling down the grade.

(b) EVat constant velocity—The EV is moving at a constant velocity along a road that has a constant grade of -12%.

(i) Plot, on the same graph, the magnitudes of the tangential gravitational force (FgxT), the aerodynamic drag force (FAD), and the rolling resistance force (Froll) versus velocity for 0<V= 180 mph. Over that range of velocity, does FgxT dominate? When does FAD dominate? When does Froll dominate? Label these regions on the graph.

(ii) Derive an expression for the tractive force as a function of velocity. Plot this expression on its own graph. Is the tractive force always in the same direction?

Showing all steps, derive and plot the velocity profile [i.e., v(t)] for constant FTR-constant grade acceleration. (Constant grade means that B is constant but not necessarily zero.) Given:

1. The EV starts from rest at t=0.

2. The resultant of FgxT and FTR is enough to overcome the rolling resistance to get the EV moving.

What effect does gravity have on the velocity profile?

A vehicle accelerates from 0 to 60 mph in 10 s with the velocity profile given by

The vehicle is on level road. For the problem, use the parameters given in Problem 2.3.

(a) Calculate and plot FTR(t) and PTR(t) for 0=t=10 s

(b) Calculate eTR. How much of eTR is KE? How much is

Eloss?

Using the vehicle parameters given in Problem 2.3, calculate and plot, on the same graph, steady state Ftrversus Vcharacteristics for B=±4° and -60 mph =V= 60 mph, but V 0.

An electric vehicle racer will attempt to jump five city buses as shown in Figure P2.7. The vehicle will start at rest at a position 100 m from the beginning of the take-off ramp. The vehicle will accelerate uniformly, until it reaches the end of the take-off ramp, at which time it will be traveling at 100 mph.

The vehicle has the following parameter values:

Also, take density of air =1.16 kg/m3, and acceleration due to gravity g=9.81 m/s2.

(a) Sketch and label the velocity profile of the vehicle from the time it starts to the time it reaches the end of the take-off ramp. How much time does the vehicle take to reach the end of the take-off ramp?

(b) Calculate the change in gravitational potential energy, from the start to the end of the take-off ramp.

(c) Calculate energy loss, Eioss, from the start to the end of the take-off ramp, if eTR=8.28x105 J during that period. FIGURE P2.7 