Runner A is running at a constant rate of 2.3 m/s when he passes RUmmer 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 analyze their respective positions as a function of time.

Let's denote the time it takes for Runner B to catch up to Runner A as "t" (in seconds).

Runner A has a constant velocity of 2.3 m/s, so the distance he covers in time "t" is given by:
Distance of Runner A = Velocity of Runner A × Time
Distance of Runner A = 2.3 m/s × t

Runner B starts running 4 seconds after Runner A has passed him. So, the time passed for Runner B is (t + 4) seconds.
Since Runner B is accelerating at a rate of 0.5 m/s^2, we can use the equation of motion to find the distance he covers in time "t + 4":
Distance of Runner B = Initial Velocity × Time + 0.5 × Acceleration × Time^2

Since Runner A and Runner B will be at the same position when they meet, we can set their respective distances equal to each other:
2.3 m/s × t = Initial Velocity × (t + 4) + 0.5 × Acceleration × (t + 4)^2

Now, we can solve this equation to find the value of "t" which represents the time it will take Runner B to catch up to Runner A.

To find out how long it will take Runner B to catch up to Runner A, we first need to determine the distance traveled by Runner A during the 4 seconds it took Runner B to start running.

The distance traveled by Runner A during those 4 seconds can be calculated by multiplying their constant velocity by the time:

Distance = Velocity × Time
Distance = 2.3 m/s × 4 s
Distance = 9.2 meters

Now, we know that Runner B starts 9.2 meters behind Runner A and is accelerating at a rate of 0.5 m/s^2. We need to determine how long it will take for Runner B to catch up, which means the distance traveled by Runner B must equal the distance traveled by Runner A.

Let's assume t is the time it takes for Runner B to catch up. During this time, Runner A will also be running.

Let's find the distance run by Runner B:

Distance run by Runner B = Initial Velocity × Time + 0.5 × Acceleration × Time^2

Since Runner B starts from rest, the initial velocity is 0 m/s:

Distance run by Runner B = 0 × t + 0.5 × 0.5 m/s^2 × t^2
Distance run by Runner B = 0.25t^2 meters

We can set this distance equal to the distance traveled by Runner A (9.2 meters):

0.25t^2 = 9.2

To solve this equation, we can divide both sides by 0.25:

t^2 = 9.2 / 0.25
t^2 = 36.8

Finally, we take the square root of both sides to find the value of t:

t = √(36.8)
t ≈ 6.07 seconds

Therefore, it will take Runner B approximately 6.07 seconds to catch up to Runner A.