On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <h3 id="back-link"><a href="/#2.1">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $2.1.28.$ On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?
    </p>

    <h3>Solution</h3>
    <p>
        <center>

1. The external force when the car is moving is the friction force $F = \mu mg$, therefore, without taking into account the resistance from the air, the dynamic equation of motion has the form $$ma = \mu mg$$

$$a = \mu g$$
2. The kinematic equations of motion in this case are represented as follows:
$$\left{\begin{matrix} v = v_0-at \ x=v_0-\frac{at^2}{2} \end{matrix}\right.$$

$$t=\frac{v_0}{a}=\frac{v_0}{\mu g}$$
3. When substituting the values ​​of acceleration and time into the second equation, we obtain the equation for the braking distance of a car
$$x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g}$$

$$v_0 = \sqrt{2\mu gx}$$
Hence the speed needs to be
reduced by $\sqrt{10}$ times

    <h4>Answer: $\sqrt{10}$ times</h4>


<footer class="row container">
  <br>
    <p>
        <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
    </p>
    <p>
        <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
    </p>
</footer>