Question

A car is moving with speed 20 m/s. Suddenly the driver sees the sign of danger at a distance of 50m; after a certain reaction time $t_0$ he applies brakes to cause deceleration$5m{s^{ - 2}}$. What is the maximum allowable reaction time $t_0$ to avoid accidents and distance travelled by the during reaction time?

Answer 1

Hint: The given problem is from rectilinear motion. To avoid an accident of a car first we calculate the minimum distance covered with deceleration of $5m{s^{ - 2}}$. After that we calculate maximum allowable reaction time to cover 50m distance.

Complete step by step answer:
The given data regarding motion is
Speed (u)= 20m/s
Distance to avoid accident (d)= 50m
Retardation(a)= $5m{s^{ - 2}}$
To avoid an accident of a car first we calculate the minimum distance covered with deceleration of $5m{s^{ - 2}}$. After that we calculate maximum allowable reaction time to cover 50m distance.

Let the possible reaction time is ${t_0}$ , So the car moves $20{t_0}$ meter due to uniform motion in the reaction time.
After that he applies brakes with retardation of 5m/s2. The distance covered with this retardation is given by the equation of motion for uniform acceleration motion.
${v^2} = {u^2} + 2as$
Due to retardation the sign of acceleration will be negative in this equation.
Here v= Final speed= Zero (to avoid accident)
u= initial speed
s= travelled distance
After putting all the values in this equation, we will get the value of travelled distance to stop the car with retardation of 5m/s2.
$0 = {20^2} + 2( - 5)$
From this, the value of s will be
$s = \dfrac{{400}}{{10}} = 40m$

So the total distance covered by car in $t_0$ time and retardation time is (20$t_0$+40) meter. This travelled distance should be less than 50m to avoid an accident with a car.
So, the value of reaction time will be
$
  20{t_0} + 40 \leqslant 50 \\
   \Rightarrow 20{t_0} \leqslant 50 - 40 \\
   \Rightarrow 20{t_0} \leqslant 10 \\
  \therefore {t_0} \leqslant 0.5\sec \\
 $
The value of reaction time will be less than 0.5 seconds to avoid an accident with the car.

Note: Sometimes the retardation of the car or deceleration of the car is taken positively in the equation of motion. We should always remember that the acceleration is a vector quantity, so the direction of it should be considered in the equation to get the right result.