Runner A is running at a constant rate of 2.3 m/s when he passes runner B who is stationary. Runner A continues running at a constant pace, and Runner B begins to run 4 seconds after Runner A just passes him. If Runner B is accelerating at a rate of .5 m/s^2, how long will it take Runner B to catch up?

To find out how long it will take Runner B to catch up to Runner A, we need to calculate the time it will take for Runner B to cover the distance between them.

First, we need to determine the distance Runner A covers in the 4 seconds until Runner B starts running. The formula to calculate distance is given by:

distance = speed × time

Runner A's speed is 2.3 m/s, and the time is 4 seconds. Therefore, the distance Runner A covers in 4 seconds is:

distance = 2.3 m/s × 4 s
distance = 9.2 m

Now we need to calculate the equation of motion for Runner B using the equation:

distance = initial velocity × time + (1/2) × acceleration × time^2

In this case, the initial velocity of Runner B is 0 m/s since he is stationary when Runner A passes him. The acceleration of Runner B is 0.5 m/s^2, and we want to find the time it takes for Runner B to catch up with Runner A.

Since Runner B starts moving when Runner A has already covered a distance of 9.2 m, the equation becomes:

9.2 m = 0 m/s × t + (1/2) × 0.5 m/s^2 × t^2

Simplifying the equation:

9.2 m = (1/2) × 0.5 m/s^2 × t^2
9.2 m = 0.25 m/s^2 × t^2

To solve for t, we can divide both sides of the equation by 0.25 m/s^2:

t^2 = (9.2 m) / (0.25 m/s^2)
t^2 = 36.8 s^2

Finally, taking the square root of both sides gives the time it will take for Runner B to catch up:

t = √(36.8 s^2)
t ≈ 6.06 s

Therefore, it will take approximately 6.06 seconds for Runner B to catch up with Runner A.