Determine the stopping distances for a car with an initial speed of 93 km/h and human reaction time of 3.0 s for the following accelerations:
-4.00 m/s^2
-8.00 m/s^2
Vo = 93 km/h = 25.8 m/s
Lat a be the acceleration.
Distance travelled during reaction time = 25.8 * (reaction time)
Distance travelled while decelerating = Vo^2/(2|a|)
Add the times.
Anyone with a reaction time of 3.0 seconds should not be driving. Recheck the numbers in the problem.
To determine the stopping distance for a car with a given acceleration, you need to consider two factors: the distance traveled during the driver's reaction time and the distance traveled while decelerating. Let's calculate the stopping distances for the given accelerations of -4.00 m/s^2 and -8.00 m/s^2.
First, let's calculate the distance traveled during the driver's reaction time. The formula for the distance traveled during the reaction time is:
d_reaction = v_initial * t_reaction
where:
- d_reaction is the distance traveled during the reaction time
- v_initial is the initial velocity of the car in meters per second (m/s)
- t_reaction is the human reaction time in seconds (s)
Given:
- v_initial = 93 km/h = (93 * 1000) / (60 * 60) m/s = 25.83 m/s
- t_reaction = 3.0 s
Plugging in the values into the formula, we have:
d_reaction = 25.83 m/s * 3.0 s = 77.49 m
So, the distance traveled during the reaction time is 77.49 meters.
Next, let's calculate the distance traveled while decelerating. The formula for the distance traveled while decelerating is derived from the kinematic equation:
d_deceleration = (v_final^2 - v_initial^2) / (2 * a)
where:
- d_deceleration is the distance traveled while decelerating
- v_final is the final velocity of the car, which is 0 m/s since it comes to a stop
- v_initial is the initial velocity of the car in meters per second (m/s)
- a is the acceleration in meters per second squared (m/s^2)
Given:
- v_final = 0 m/s
- v_initial = 25.83 m/s
- a = -4.00 m/s^2 (for the first case) and -8.00 m/s^2 (for the second case)
Let's calculate the distances traveled while decelerating for both cases:
d_deceleration_case1 = (0^2 - 25.83^2) / (2 * -4.00)
d_deceleration_case2 = (0^2 - 25.83^2) / (2 * -8.00)
Calculating each:
d_deceleration_case1 = (-665.91) / (-8.00) = 83.24 m
d_deceleration_case2 = (-665.91) / (-16.00) = 41.62 m
So, for an acceleration of -4.00 m/s^2, the distance traveled while decelerating is 83.24 meters. For an acceleration of -8.00 m/s^2, the distance traveled while decelerating is 41.62 meters.
Finally, to calculate the total stopping distance, we sum the distance traveled during the reaction time and the distance traveled while decelerating:
stopping_distance_case1 = d_reaction + d_deceleration_case1
stopping_distance_case2 = d_reaction + d_deceleration_case2
Calculating each:
stopping_distance_case1 = 77.49 m + 83.24 m = 160.73 m
stopping_distance_case2 = 77.49 m + 41.62 m = 119.11 m
So, for an acceleration of -4.00 m/s^2, the stopping distance is 160.73 meters. For an acceleration of -8.00 m/s^2, the stopping distance is 119.11 meters.