a typical automobile has a maximum deceleration of about 7m/s2, the typical reaction time to engage the brakes is 0.50s. A school board sets the speed limit in a school zone to meet the condition that all cars should be able to stop in a distance of 4 m. a) what maximum speed should be allowed for a typical automobile? b) What fraction of the 4 m is due to the reaction time?
a)
Reaction time distance + Deceleration distance = 4 m
Vmax*(0.5s) + [(Vmax)/2]*Vmax/a = 4 m
0.5 Vmax + Vmax^2/14 = 4
Vmax = 4.78 m/s = 17.2 km/h = 10.7 mph
To find the maximum speed limit for a typical automobile in a school zone, we need to consider the deceleration and reaction time.
a) To determine the maximum speed, we can use the equation of motion:
vf^2 = vi^2 + 2ad
where:
- vf is the final velocity (0 m/s as the car needs to stop)
- vi is the initial velocity (the maximum speed we want to determine)
- a is the deceleration (-7 m/s^2, as it is the maximum deceleration)
- d is the distance the car needs to stop (4 m)
Rearranging the equation, we have:
vi^2 = vf^2 - 2ad
Since vf is 0 m/s, the equation becomes:
vi^2 = -2ad
Substituting the given values, we can solve for vi:
vi^2 = -2(-7 m/s^2)(4 m)
vi^2 = 56 m^2/s^2
vi ≈ √(56) m/s
vi ≈ 7.48 m/s
Therefore, the maximum allowable speed for a typical automobile should be approximately 7.48 m/s (or rounded to 7.5 m/s).
b) To determine the fraction of the 4 m distance that is due to the reaction time, we can use the formula:
distance due to reaction time = speed × reaction time
Given that the reaction time is 0.50 s, we can calculate the distance:
distance due to reaction time = 7.48 m/s × 0.50 s
distance due to reaction time ≈ 3.74 m
Therefore, approximately 3.74 m of the 4 m total stopping distance is due to the reaction time.