A hawk is flying horizontally at 14.0 m/s in a straight line, 250 m above the ground. A mouse it has been carrying struggles free from its grasp. The hawk continues on its path at the same speed for 2.00 s before attempting to retrieve its prey. To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3.00 m above the ground.
(a) Assuming no air resistance, find the diving speed of the hawk.
To find the diving speed of the hawk, we can use the kinematic equations of motion. Let's break down the problem step by step.
Step 1: Calculate the time it took for the hawk to dive.
To find the diving speed, we need to know the time it took for the hawk to dive. We know it continued flying horizontally at the same speed for 2.00 seconds. Therefore, the time it took for the hawk to dive is the time it took to travel a horizontal distance of 250 meters.
Using the formula: distance = speed × time, we can rearrange the formula to solve for time:
time = distance / speed
In this case, the distance is 250 meters, and the speed is 14.0 m/s. Substituting those values into the formula, we get:
time = 250 m / 14.0 m/s
time ≈ 17.86 seconds (rounded to 2 decimal places)
Step 2: Calculate the diving speed of the hawk.
To calculate the diving speed of the hawk, we'll use the concept of relative velocity. Since the hawk traveled horizontally while diving, its horizontal velocity remains the same as its initial speed, which is 14.0 m/s.
The height from where the hawk dived down to capture the mouse, 250 m - 3.00 m = 247.00 m.
Using the formula: distance = speed × time, we can rearrange it to solve for speed:
speed = distance / time
In this case, the distance is 247.00 meters, and the time is 17.86 seconds (as calculated earlier). Substituting those values into the formula, we get:
speed = 247.00 m / 17.86 s
speed ≈ 13.82 m/s (rounded to 2 decimal places)
Therefore, the diving speed of the hawk is approximately 13.82 m/s.