Chapter 5.2, Problem 25PP

To determine

Whether the car stop before hitting the branch of a tree which is lying on the road.

Answer to Problem 25PP

No

Explanation of Solution

Given:

Initial velocity of the car, $v_i=23\text{ m/s}$

Distance to the tree branch lying on the road $=60.0\text{ m}$

Coefficient of kinetic friction, ${\mu }_k=0.41$

Mass of the car, $m=1200\text{ kg}$

Formula used:

Let the car is moving in positive x-direction.

According to Newton’s law, the friction force exerted by the road on the car must be equal and opposite to the force (brake) applied on the car by the driver.

That is, ${\text{F}}_{\text{net}}=-F_{f,\text{kinetic}}$$\left(1\right)$

Where, $F_{f,\text{kinetic}}$ is kinetic friction force.

  $F_{f,\text{kinetic}}={\mu }_k{F}_N$

Substituting for $F_{f,\text{kinetic}}$ in equation $\left(1\right)$ ,

  ${\text{F}}_{\text{net}}=-{\mu }_k{F}_N$$\left(2\right)$

Where, $F_N$ is the normal force, $F_N=mg$

Substituting ${\text{F}}_{\text{net}}=ma$ and $F_N=mg$ in equation $\left(2\right)$ ,

  $ma=-{\mu }_{k}mg$

  $\Rightarrow a=-{\mu }_{k}g$

Where, $a$ is acceleration of the car and $g$ is acceleration due to gravity, $g=9.8{\text{ m/s}}^2$

Now to find the distance, use the equation of motion,

  $v_f^2=v_i^2+2a\left(d_f-d_i\right)$$\left(3\right)$

Where, $v_f$ is the final velocity of the car, $v_f=0\text{ m/s}$

Let $d_i=0$ and solve for $d_f$

Substituting for $v_f$ and $d_i$ in equation $\left(3\right)$ ,

  $0=v_i^2+2a\left(d_f-0\right)$

  $v_i^2+2a{d}_f=0$

  $\Rightarrow 2a{d}_f=-v_i^2$

  $\Rightarrow d_f=\frac{-v_i^2}{2a}$

Substituting $a=-{\mu }_{k}g$ in the above equation,

  $\Rightarrow d_f=\frac{-v_i^2}{2\left(-{\mu }_{k}g\right)}$$\left(4\right)$

Calculation:

Substituting the numerical values in equation $\left(4\right)$ ,

  $d_f=\frac{-{\left(23\text{ m/s}\right)}^2}{2\left(-0.41\right)\left(9.8{\text{ m/s}}^2\right)}$

  $=\frac{-529{\text{ m}}^2/{\text{s}}^2}{-8.036{\text{ m/s}}^2}$

  $=65.8\text{ m}$

Thus,he hits the branch before he can stop the car.

Conclusion:

No. Car will not stop before hitting the branch.