The speed limit in a school zone is 40km/h. A driver traveling at this speed sees a child run into the road 13m ahead of his car. He applies the breaks, and the car decelerates at uniform rate of 0.8m/s². If the driver's reaction times is 0.25s, will the car stop before hitting the child?
Given:
Vo = 40km/h = 11.1 m/s.
d1 = 13m. = Stopping distance.
a = -8.0 m/s^2?.
t = 0.25s. = Reaction time.
d2 = V*t =11.1 * 0.25 = 2.78 m.
d = d1-d2 = 13 - 2.78 = 10.2 m = Required stopping distance.
V^2 = Vo^2 ^ 2a*d = 0.
11.1^2 - 5.56*d = 0,
d = 22.2 m = Stopping distance.
40 km/h * 1000/3600 = 11.1 m/s initial Vi
goes for .25 sec *11.1 = 2.78 m before braking
average speed = 11.1/2 = 5.55 m/s
v = Vi + a t
a = -0.8
0 = 11.1 -0.8 t
t = 13.9 s (I have a hunch your a should be -8.0 m/s^2 not -0.8)
then d = Vi t + (1/2) a t^2
d = 11.1(13.9) - 0.4 (13.9)^2
d = 154 - 77.3 = 76.7 meters
76.7 + initial 2.78 = 79.5 meters
Well, it seems like this driver has found themselves in quite a pickle, huh? Let's break it down and crunch some numbers to see what might happen!
First, we need to find out the initial speed of the car when the driver spots the child. The reaction time of the driver is 0.25 seconds, so during that time, the car is still moving at its original speed of 40 km/h.
Now, we need to convert 40 km/h to m/s. Since 1 km/h is equal to 0.27778 m/s, we can multiply 40 by 0.27778 to get approximately 11.111 m/s.
The car is moving at 11.111 m/s when the driver sees the child 13m ahead. The driver then starts applying the brakes, and the car decelerates at a uniform rate of 0.8 m/s².
To find out if the car will stop before hitting the child, we need to calculate the distance it will take to stop. We can use the equation: v² = u² + 2as, where v is the final velocity (which is 0 m/s since the car stops), u is the initial velocity (11.111 m/s), a is the acceleration (-0.8 m/s²), and s is the distance to be covered (unknown).
Rearranging this equation, we get s = (v² - u²) / (2a).
Plugging in the values, we get s = (0 - (11.111)²) / (2 * (-0.8)), which simplifies to s = 154 / 1.6, approximately equal to 96.25 meters.
So, if the car is able to stop within 96.25 meters after the driver sees the child, the child will be safe. Since the distance between the car and the child is only 13 meters, clown logic suggests that the car will indeed stop in time and avoid hitting the child!
Remember, my answers come with a twist of humor, so please don't take them too seriously. Safety is no laughing matter, so please drive responsibly and make sure to follow all rules and speed limits!
To determine if the car will stop before hitting the child, we need to calculate the stopping distance of the car and compare it to the distance between the car and the child.
First, let's calculate the stopping distance of the car. The stopping distance comprises two parts: the reaction distance (the distance the car travels during the driver's reaction time) and the braking distance (the distance the car travels while decelerating).
1. Reaction Distance:
The driver's reaction time is given as 0.25 seconds. During this time, the car will continue to travel at its initial speed. We can calculate the reaction distance using the formula:
Reaction distance = initial speed × reaction time
The initial speed is 40 km/h, which needs to be converted to meters per second:
40 km/h = (40 × 1000) m / (60 × 60) s = 11.11 m/s
Reaction distance = 11.11 m/s × 0.25 s = 2.78 meters
2. Braking Distance:
To calculate the braking distance, we need to use the equation of motion:
Final velocity² = Initial velocity² + 2 × acceleration × distance
The final velocity is 0 m/s because the car is supposed to stop. The initial velocity is the same as the initial speed, which is 11.11 m/s. The acceleration is given as -0.8 m/s² (negative because it represents deceleration). We can plug in these values and solve for the distance:
0 = (11.11 m/s)² + 2 × (-0.8 m/s²) × distance
distance = [(11.11 m/s)²] / [2 × 0.8 m/s²] = 76.94 meters
Now, the total stopping distance is the sum of the reaction distance and the braking distance:
stopping distance = reaction distance + braking distance
stopping distance = 2.78 meters + 76.94 meters
stopping distance = 79.72 meters
Since the stopping distance is greater than the actual distance between the car and the child (13 meters), we can conclude that the car will stop before hitting the child, and the driver will successfully avoid the collision.