To figure this out, you can do the following:
Step 1: Define your Constants
Let's use a typical dart mass of m = 0.03 kg, a distance from the floor to the dart's initial position s = 2 meters, and the acceleration due to gravity g = 9.8 m/s^2.
Step 2: Calculate the Time it Takes for the Dart to Hit the Floor
You can use the equation of motion for free fall which is s = ut + 1/2 * g * t^2 where s is the distance travelled, u is the initial speed, g is the acceleration, and t is the time. Since the dart is released from rest, u = 0, so the equation simplifies to s = 1/2 * g * t^2. From this, you can find the time t it takes for the dart to fall by rearranging the equation to solve for t: t = sqrt(2 * s / g). Substituting the known values in, we get t = sqrt(2 * 2 / 9.8) = 0.6388765649999398 seconds.
Step 3: Determine the Root Mean Square Distance by Which the Dart will Miss the Crack
This is determined by the standard deviation of a uniform distribution between -s and s, which can be interpreted as the average distance of a random point from the mean in a uniform distribution. This is given by sqrt(s^2/3). Substituting in the known values, we get sqrt(2^2 / 3) = 1.1547005383792515 m.
In conclusion, when releasing the dart from a height of 2 meters, it will take approximately 0.64 seconds to hit the floor and the average miss distance for the dart will be about 1.15 meters.