A driver has a reaction time of 0.50 s, and the maximum deceleration of her car is 6.0 m/s squared. She is driving at 20 m/s when suddenly she sees an obstacle in the road 50 m in front of her. Can she stop the car to avoid the collision?
in that first half second she goes 10 meters, 40 meters left
then
v = 20 - 6 t
at stop v = 0 so
t = 20/6
average speed during stop = (20 + 0) /2 = 10 m/s
so goes 10 *20/6 more = 100/3 = 33 1/3 meters
well, we had 40 :)
Well, let me crunch some numbers while riding my tiny tricycle of knowledge here. The obstacle is 50 meters in front of her, and her maximum deceleration is 6.0 m/s². So, to figure out if she can stop in time, let's calculate the distance she can cover during her reaction time.
Given that her reaction time is 0.50 s, and she's traveling at 20 m/s, we multiply the two to find: (0.50 s) x (20 m/s) = 10 meters. The reaction distance alone is 10 meters.
Now let's combine her reaction distance with her maximum deceleration. If she slams the brakes right after spotting the obstacle, she'll be decelerating at 6.0 m/s². Assuming she's already traveled 10 meters during her reaction time, she has 40 meters left to stop.
Using the equation of motion, s = (v² - u²) / (2a), we can solve for the stopping distance (s). Here, v = 0 m/s (her final velocity), u = 20 m/s (her initial velocity), and a = -6.0 m/s² (the negative sign is because she's decelerating).
Plugging in those values, we get: s = (0 - 20²) / (2 x -6.0).
Calculating further, she'll need a stopping distance of 66.67 meters. However, she only has 40 meters available before the obstacle. So, unfortunately, she won't be able to stop in time. Looks like her car and that obstacle might become really good friends.
To determine if the driver can stop the car to avoid the collision, we need to calculate the distance that the car will travel during the driver's reaction time.
Given:
Reaction time (t) = 0.50 s
Initial velocity (u) = 20 m/s
Deceleration (a) = -6.0 m/s^2 (negative sign indicates deceleration)
Distance to the obstacle (s) = 50 m
Step 1: Calculate the distance traveled during the reaction time using the formula: s = ut
Distance (s1) = 20 m/s × 0.50 s = 10 m
Step 2: Calculate the remaining distance (s2) to be covered to avoid the collision.
Remaining distance (s2) = Total distance - Distance traveled during the reaction time
s2 = 50 m - 10 m = 40 m
Step 3: Use the equation of motion to calculate the distance to stop the car:
v^2 = u^2 + 2as
0^2 = (20 m/s)^2 + 2(-6.0 m/s^2)s
-240s = -400
s = -400 / -240
s = 1.67 m
Therefore, the car will stop within 1.67 meters, which is less than the remaining distance of 40 meters. So, the driver will not be able to stop the car in time to avoid the collision.
To determine if the driver can stop the car to avoid the collision, we need to calculate two things: the time it will take for the driver to react and apply the brakes (reaction time), and the time it will take for the car to come to a stop.
First, let's calculate the time it takes for the driver to react and apply the brakes. The reaction time given is 0.50 seconds.
Next, let's calculate the time it will take for the car to come to a stop. We can use the following equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, as the car needs to come to a stop)
u = initial velocity (20 m/s)
a = acceleration or deceleration (-6.0 m/s^2, as it is the maximum deceleration)
s = distance (50 m)
Rearranging the equation, we have:
s = (v^2 - u^2) / (2a)
Substituting the given values into the equation, we get:
s = (0 - (20^2)) / (2 * (-6.0))
s = (-400) / (-12)
s = 33.33 m
Therefore, it will take a distance of 33.33 meters for the car to come to a stop.
Now, let's calculate the total distance covered during the driver's reaction time:
d = v * t
d = 20 * 0.50
d = 10 m
So, during the driver's reaction time of 0.50 seconds, the car would cover a distance of 10 meters.
To check if the driver can stop the car to avoid the collision, we need to determine if the total distance covered during the reaction time (10 m) plus the distance required to stop the car (33.33 m) is greater than the distance to the obstacle (50 m).
Total distance = distance covered during reaction time + distance required to stop
Total distance = 10 m + 33.33 m
Total distance = 43.33 m
Since the total distance (43.33 m) is less than the distance to the obstacle (50 m), the car will not be able to stop in time to avoid the collision.