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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $2.1.28.$ 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? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| Forces acting on the car | | Forces acting on the car |
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| 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 | | 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 |
| $$ ma = \mu mg $$ | | $$ ma = \mu mg $$ |
| | | |
| $$ a = \mu g $$ | | $$ a = \mu g $$ |
| 2. The kinematic equations of motion in this case are represented as follows: | | 2. The kinematic equations of motion in this case are represented as follows: |
| $$ \left\{\begin{matrix} | | $$ \left\{\begin{matrix} |
| v = v_0-at \\ | | v = v_0-at \\ |
| x=v_0-\frac{at^2}{2} | | x=v_0-\frac{at^2}{2} |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| | | |
| $$ t=\frac{v_0}{a}=\frac{v_0}{\mu g} $$ | | $$ t=\frac{v_0}{a}=\frac{v_0}{\mu g} $$ |
| 3. When substituting the values of acceleration and time into the second equation, we obtain the equation for the braking distance of a car | | 3. When substituting the values of acceleration and time into the second equation, we obtain the equation for the braking distance of a car |
| $$ x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g} $$ | | $$ x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g} $$ |
| | | |
| $$ v_0 = \sqrt{2\mu gx} $$ | | $$ v_0 = \sqrt{2\mu gx} $$ |
| Hence the speed needs to be | | Hence the speed needs to be |
| reduced by $\sqrt{10}$ times | | reduced by $\sqrt{10}$ times |
| </p> | | </p> |
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| <h4>Answer: $\sqrt{10}$ times</h4> | | <h4>Answer: $\sqrt{10}$ times</h4> |
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