PRACTICE

ENGINE$-$GENERATED TRACTIVE EFFORT AND THEORETICAL

PROBLEM $2\mathrm{.}3$ MINIMUM STOPPING DISTANCE

A car has $11-$inch radius wheels and is traveling up a $3\%$ grade with a gear reduction ratio of $2$ to $1$ and a driveline slippage of $2\%.$ The car has a torque$/$engine speed curve that is given by the equation,

$M_e=7n_c-0\mathrm{.}08n_e^2$

with $M_e$ in $ft-lb$ and $n_e$ in revolutions per second. It is also known that the car is traveling at a speed where its engine is generating maximum torque. If the car is on good, wet pavement with antilock brakes and a $95\%$ braking efficiency, ignoring air resistance, what would be the theoretical stopping distance $($distance traveled until a stop after the brakes are applied$)?$

SOLUTION Note: Open boxes in equations $$ 'are to be completed by the reader

To solve this problem we must first determine the speed at which the vehicle was traveling before the brakes were applied. The problem states that the car was traveling is traveling at a speed that has its engine producing maximum torque. So we need to find the engine speed at this maximum torque, and then use this engine speed along

$2\mathrm{.}10$ Practice Problems

$49$

with the given gear reduction ratio and driveline slippage to arrive at vehicle speed $($by applying Eq$.2\mathrm{.}18).$

As in Practice Problem $2\mathrm{.}1,$ finding the engine speed $n_e$ at maximum torque is achieved by taking the first derivative of $M_e=7n_e-0\mathrm{.}08n_e^2$ with respect to engine speed $n_e$ and setting it equal to zero $($to get the maximum$)$:

$\frac{d{M}_e}{d{n}_e}=7-,k{n}_e=0$

$n_e=\frac{7}{}=43\mathrm{.}75$ revolutions per second

With the engine speed, Eq$.2\mathrm{.}18$ is applied as follows,

$V=\frac{2r{n}_e(1-i)}{{}_0}$

$V=\frac{\frac{}{}(1-)}{}=123\mathrm{.}45f\frac{t}{s}$

Eq$.2\mathrm{.}43$ can now be applied to determine the theoretical stopping distance,

$S=\frac{{}_b(V_1^2-V_2^2)}{2g({}_b+f_n+-sin{}_R)}$

In the above equation, ${}_b$ and $sin{}_R$ are given in the problem, ${}_b$ is a known constant $($see text$),$ can be obtained from Table $2\mathrm{.}5,$ and $f_{rl}$ can be determined from Eq$.2\mathrm{.}5$ as$,$

So Eq$.2\mathrm{.}43$ is solved as$,$

$s=\frac{({}^2-{}^2)}{2(++-)}=273,69n_{{?}_{?}}.$