A cheetah is hunting. Its prey runs for 3.07 s at a constant velocity of +11.32 m/s. Starting from rest, what constant acceleration must the cheetah maintain in order to run the same distance as its prey runs in the same time?
To find the constant acceleration required by the cheetah to cover the same distance as its prey in the same time, we can use the kinematic equation:
distance = initial velocity * time + (1/2) * acceleration * (time^2)
First, let's calculate the distance covered by the prey using the given information:
distance_prey = velocity_prey * time
distance_prey = 11.32 m/s * 3.07 s
distance_prey = 34.7264 m
Now, we'll set up the equation to find the acceleration required by the cheetah:
distance_cheetah = 0 * t + (1/2) * acceleration * (t^2)
distance_cheetah = (1/2) * acceleration * (t^2)
distance_cheetah = (1/2) * acceleration * (3.07s)^2
Since we want the cheetah to cover the same distance as the prey, distance_cheetah = distance_prey. We can substitute the values into the equation:
34.7264 m = (1/2) * acceleration * (3.07s)^2
Now, let's solve for the acceleration:
acceleration = (34.7264 m) / [(1/2) * (3.07s)^2]
acceleration = 34.7264 m / [(1/2) * 9.4249 s^2]
acceleration = 34.7264 m / 4.71245 s^2
acceleration ≈ 7.37 m/s^2
Therefore, the cheetah must maintain a constant acceleration of approximately 7.37 m/s^2 to cover the same distance as its prey in the same time.