A rabbit and a fox ran a race. They started in the same direction, from the same place. The rabbit accelerated 5.0m/s2 up to its maximum speed. It...

To determine the distance traveled by the rabbit in 8 seconds, let's divide it into two time intervals. The time intervals are from 0 to 4 seconds and from 4 to 8 seconds.

At time interval from t=0s to t=4s, the rabbit starts from rest and has a constant acceleration of 5m/s^2. Applying the formula

`d= v_i Delta t + 1/2a Delta t ^2`

it traveled a distance of

`d_(R1) = 0 + 1/2 (5)(4)^2`...

To determine the distance traveled by the rabbit in 8 seconds, let's divide it into two time intervals. The time intervals are from 0 to 4 seconds and from 4 to 8 seconds.

At time interval from t=0s to t=4s, the rabbit starts from rest and has a constant acceleration of 5m/s^2. Applying the formula

`d= v_i Delta t + 1/2a Delta t ^2`

it traveled a distance of

`d_(R1) = 0 + 1/2 (5)(4)^2`  

`d_(R1) = 40`  meters

Next, applying the formula

`v_f = v_i + a Delta t`

the maximum speed of the rabbit is:

`v_(R) = 0 + 5*4`

`v_(R) = 20` m/s

At the time interval from t=4s to t=8s, the rabbit is running at a constant speed v_(R). So, its acceleration is zero. It has traveled a distance of

`d_(R2) = v_(R) Delta t +1/2a Delta t^2`

`d_(R2) = 20 * 4 + 0`

`d_(R2) = 80m`

Hence, the rabbit has traveled a total distance of

`d_R= d_(R1) + d_(R2)`

`d_R = 40 + 80`

`d_R = 120 m`

To determine the distance traveled by the fox, apply the formula:

`d= v_i Delta t + 1/2at^2`

Since the fox starts from rest and accelerates at 2 m/s^2, the distance traveled from t=0s to t=8s is:

`d_F = 0 + 1/2*2 * 8^2`

`d_F = 64` meters

The distance between the rabbit and the fox at t=8s is:

`d = d_R - d_f`

`d = 120 - 64`

`d=56 m`

Therefore, at t=8s, the rabbit is ahead by 56 meters.