A cheetah is hunting. Its prey runs for 3.21 s at a constant velocity of +8.36 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?
The cheetah must have the same average velocity during the interval. Since it accelerates uniformly from zero, its maximum velocity must be twice the average, or 16.72 m/s. The acceleration rate is
(16.72 m/s)/(3.21 s) = ___ m/s^2
To determine the constant acceleration the cheetah must maintain in order to cover the same distance as its prey, we can start by calculating the distance traveled by the prey using the formula:
Distance = Velocity * Time
Given:
Velocity of the prey = +8.36 m/s
Time = 3.21 s
Substituting the values into the formula:
Distance = 8.36 m/s * 3.21 s
Calculating:
Distance = 26.8516 m (approx.)
Now, we know that the cheetah needs to cover the same distance in the same time (3.21 s). Therefore, we can use the equation of motion to calculate the acceleration required. The equation of motion is:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Since the cheetah starts from rest (initial velocity = 0 m/s), we can simplify the equation:
Distance = (1/2) * Acceleration * Time^2
Substituting the values:
26.8516 m = (1/2) * Acceleration * (3.21 s)^2
Simplifying further:
53.7032 m = Acceleration * 10.3041 s^2
Dividing both sides by 10.3041 s^2:
Acceleration = 53.7032 m / 10.3041 s^2
Calculating:
Acceleration ≈ 5.209 m/s^2
Therefore, the cheetah must maintain a constant acceleration of approximately 5.209 m/s^2 to cover the same distance as its prey in the given time.