The determined Wile E. Coyote is out once more to try to capture the elusive roadrunner. The coyote wears a new pair of Acme power roller skates, which provide constant horizontal acceleration of 15 m/s^2. The coyote starts off at rest 70 m from the edge of the cliff. (a) If the roadrunner moves with constant speed, find the minimum speed the roadrunner must have to reach the cliff before the coyote. (b) If the cliff is 100 m above the base of a canyon, find where the coyote lands in the canyon. (Assume his skates are still in operation when he is in "flight" and that his horizontal component of acceleration remains constant at 15 m/s^2.)
The determined Wile E. Coyote is out once more to try to capture the elusive roadrunner. The coyote wears a new pair of power roller skates, which provide a constant horizontal acceleration of 15 m/s2. The coyote starts off at rest 70 m from the edge of a cliff at the instant the roadrunner zips by in the direction of the cliff.
To solve this problem, we'll break it down into two parts: (a) finding the minimum speed of the roadrunner, and (b) finding where the coyote lands in the canyon.
(a) Finding the minimum speed of the roadrunner:
To find the minimum speed required for the roadrunner to reach the cliff before the coyote, we need to consider the distance the roadrunner needs to cover and the time it takes for the coyote to reach the cliff.
The roadrunner needs to cover a distance of 70 m, while the coyote has a constant horizontal acceleration of 15 m/s^2. The time it takes for the coyote to reach the cliff can be found using the formula:
distance = initial velocity * time + (1/2) * acceleration * time^2
Here, the initial velocity is 0 m/s, the distance is 70 m, and the acceleration is 15 m/s^2. Solving for time:
70 = 0 * t + (1/2) * 15 * t^2
70 = (1/2) * 15 * t^2
70 = 7.5 * t^2
t^2 = 70 / 7.5
t^2 = 9.33
t ≈ 3.05 seconds
Now, we know that the speed of the roadrunner is approximately the distance (70 m) divided by the time (3.05 seconds):
speed = distance / time
speed = 70 m / 3.05 s
speed ≈ 22.95 m/s
Therefore, the minimum speed the roadrunner must have to reach the cliff before the coyote is approximately 22.95 m/s.
(b) Finding where the coyote lands in the canyon:
To find where the coyote lands in the canyon, we need to calculate the time it takes for the coyote to reach the cliff and use that time to determine the vertical distance he falls.
Since the cliff is 100 m above the base of the canyon, the total distance the coyote falls is 100 m. The time it takes for the coyote to reach the cliff, as we calculated in part (a), is approximately 3.05 seconds.
We can use the following kinematic equation to calculate the vertical distance the coyote falls:
distance = initial velocity * time + (1/2) * acceleration * time^2
Here, the initial velocity is 0 m/s, the distance is 100 m, and the acceleration is due to gravity, approximately 9.8 m/s^2. Solving for distance:
100 = 0 * t + (1/2) * 9.8 * t^2
100 = 4.9 * t^2
t^2 = 100 / 4.9
t^2 ≈ 20.41
t ≈ 4.52 seconds
Therefore, the coyote lands in the canyon after approximately 4.52 seconds. The horizontal distance the coyote travels during this time can be found using:
distance = initial velocity * time + (1/2) * acceleration * time^2
Here, the initial velocity is 0 m/s, the acceleration is 15 m/s^2, and the time is approximately 4.52 seconds. Solving for distance:
distance = 0 * 4.52 + (1/2) * 15 * (4.52)^2
distance ≈ 153.43 m
Therefore, the coyote lands in the canyon approximately 153.43 m from the base of the canyon.
In summary:
(a) The minimum speed the roadrunner must have to reach the cliff before the coyote is approximately 22.95 m/s.
(b) The coyote lands in the canyon approximately 153.43 m from the base of the canyon.