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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`  


`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.

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