Chapter 7, Problem 79P

To determine

Whether or not any of the cars was speeding.

Answer to Problem 79P

The car with mass $1100\text{kg}$ crossed the speed limit.

Explanation of Solution

The mass of the car moving west is $1100\text{kg}$. The mass of the car moving north is $1300\text{kg}$. After the collision the car got stuck and move $17\text{m}$ in a direction $30°$ north of west. The coefficient of kinetic friction between the tire and road is $0.80$. The speed limit is $70\text{km/h}$.

The figure 1 below shows the initial and final velocities of the cars.

The collision between the cars is perfectly inelastic. The final velocity of both the cars will be same. The final velocity of the collision will be the initial velocity of the sliding motion of the two cars stuck to each other. At the end of sliding, the system comes to rest.

Write the energy formula for the sliding motion of the two cars system when they are joined to each other.

$\frac{1}{2}m{v}_i^2+fd\cos 180°=0$ (I)

Here, $m$ is the combined mass of the two cars, $v_i$ is the initial speed of sliding, $f$ is the friction force, $d$ is the distance travelled before coming to rest.

Write the formula for the friction force.

$f={\mu }_{k}mg$ (II)

Here, ${\mu }_k$ is the coefficient of kinetic friction, $g$ is the acceleration due to gravity.

Substitute equation (II) in equation (I).

$\frac{1}{2}m{v}_i^2-{\mu }_{k}mgd=0$

Re-write the above equation to get an expression for $v_i$

$\begin{array}{c}\frac{1}{2}m{v}_i^2={\mu }_{k}mgd\\ v_i=\sqrt{2{\mu }_{k}gd}\end{array}$

Thus write the formula for the final speed of the collision.

$v_f=\sqrt{2{\mu }_{k}gd}$ (III)

Here, $v_f$ is the final speed of collision.

Refer figure 1 and write the equation for the conservation of linear momentum in x-direction.

$\begin{array}{c}{m}_1{v}_{i1}=\left(m_1+m_2\right)v_f\cos 150°\\ =\left(m_1+m_2\right)\left(\sqrt{2{\mu }_{k}gd}\right)\cos 150°\end{array}$

Here, $m_1$ is the mass of the first car, $m_2$ is the mass of the second car, $v_{i1}$ is the initial velocity of the first car.

Re-write the above equation to get an expression for $v_{i1}$.

$v_{i1}=\left(\frac{m_1+m_2}{m_1}\right)\left(\sqrt{2{\mu }_{k}gd}\right)\cos 150°$ (IV)

Refer figure 1 and write the equation for the conservation of linear momentum in y-direction.

$\begin{array}{c}{m}_2{v}_{i2}=\left(m_1+m_2\right)v_f\sin 150°\\ =\left(m_1+m_2\right)\left(\sqrt{2{\mu }_{k}gd}\right)\sin 150°\end{array}$

Here, $m_2$ is the mass of the second car, $v_{i2}$ is the initial velocity of the second car.

Re-write the above equation to get an expression for $v_{i2}$.

$v_{i2}=\left(\frac{m_1+m_2}{m_2}\right)\left(\sqrt{2{\mu }_{k}gd}\right)\sin 150°$ (V)

Conclusion:

Substitute $1100\text{kg}$ for $m_1$, $1300\text{kg}$ for $m_2$, $17\text{m}$ for $d$, $0.80$ for ${\mu }_k$, $9.80{\text{m/s}}^{\text{2}}$ for $g$ in equation (IV).

$\begin{array}{c}{v}_{i1}=\left(\frac{1100\text{kg}+1300\text{kg}}{1100\text{kg}}\right)\left(\sqrt{2\left(0.80\right)\left(9.80{\text{m/s}}^{\text{2}}\right)\left(17\text{m}\right)}\right)\cos 150°\\ =\left(\frac{1100\text{kg}+1300\text{kg}}{1100\text{kg}}\right)\left(\sqrt{2\left(0.80\right)\left(9.80{\text{m/s}}^{\text{2}}\right)\left(17\text{m}\right)}\right)\left(\frac{1\text{km/h}}{0.2778\text{m/s}}\right)\cos 150°\\ =-110\text{km/h}\end{array}$

Substitute $1100\text{kg}$ for $m_1$, $1300\text{kg}$ for $m_2$, $17\text{m}$ for $d$, $0.80$ for ${\mu }_k$, $9.80{\text{m/s}}^{\text{2}}$ for $g$ in equation (V).

$\begin{array}{c}{v}_{i2}=\left(\frac{1100\text{kg}+1300\text{kg}}{1300\text{kg}}\right)\left(\sqrt{2\left(0.80\right)\left(9.80{\text{m/s}}^{\text{2}}\right)\left(17\text{m}\right)}\right)\sin 150°\\ =\left(\frac{1100\text{kg}+1300\text{kg}}{1300\text{kg}}\right)\left(\sqrt{2\left(0.80\right)\left(9.80{\text{m/s}}^{\text{2}}\right)\left(17\text{m}\right)}\right)\left(\frac{1\text{km/h}}{0.2778\text{m/s}}\right)\sin 150°\\ =54\text{km/h}\end{array}$

The velocity of car with mass $1100\text{kg}$ is $-110\text{km/h}$. The velocity of the car with mass $1300\text{kg}$ is $54\text{km/h}$. Thus the car with mass $1100\text{kg}$ crossed the speed limit.