a small airplane is dropping a probe in a remote area in antarctica in a narrowly specified location. the probe in a protective shell can not land any harder than with a maximum speed of 60m/sec at landing, while the plane has to move at a minimum speed of 30m/sec horizontally in order to stay airborne.
if you are the pilot, you face the problem of having to decide from which height and distance from the location you can release the probe without destroying it . FIND A REASONABLE ANSWER FOR THIS PROBLEM
well, the vertical speed is 9.8t so the probe cannot fall for more than 60/9.8 = 6.12 seconds.
That means the plane cannot be higher than 183.67 m high.
so, at h meters altitude, with horizontal velocity v, the probe will land with speed
s = ā(v^2+4h^2)
The probe will travel horizontally vt = vā(h/4.9) meters
So, we need
h <= 183.67
v >= 30
ā(v^2+4h^2) <= 60
So, just pick a v and an h to fit those constraints.
To find a reasonable answer for this problem, we need to consider a few key factors:
1. Maximum landing speed: The probe should not land with a speed higher than 60 m/sec to ensure its protective shell remains intact.
2. Minimum horizontal speed of the airplane: The plane needs to maintain a minimum speed of 30 m/sec horizontally to stay airborne.
3. Distance from the landing location: We must determine the distance from the location where the probe needs to be dropped.
4. Height of release: We need to calculate the suitable height from which the probe can be released without exceeding the maximum landing speed.
To solve this problem, we can use basic physics equations. Let's break it down step by step:
Step 1: Calculate the time it takes for the plane to reach the landing location.
We can use the formula: time = distance / speed
Given the minimum horizontal speed (30 m/sec) and the target distance, we'll assume a value of 2,000 meters for demonstration purposes:
time = 2000 m / 30 m/sec = 66.67 sec
Step 2: Determine the acceleration due to gravity.
The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s^2.
Step 3: Calculate the maximum height from which the probe can be released.
We'll assume an initial vertical speed of 0 m/sec when the probe is dropped. Using the formula: height = (1/2) * g * t^2
height = (1/2) * 9.8 m/s^2 * (66.67 sec)^2 = 21778.90 meters
Step 4: Adjust the height considering the maximum landing speed of the probe.
We need to find out how much the probe's velocity will increase due to gravity during the fall. Using the formula: velocity = g * time
velocity = 9.8 m/s^2 * 66.67 sec = 654.17 m/sec
To ensure the probe does not exceed the maximum landing speed, we subtract this velocity from the maximum landing speed:
allowed vertical velocity = 60 m/sec - 654.17 m/sec = -594.17 m/sec
Step 5: Calculate the revised maximum height accounting for the allowed vertical velocity.
Using the formula: height = (1/2) * g * t^2 + (allowed vertical velocity)*t
height = (1/2) * 9.8 m/s^2 * (66.67 sec)^2 + (-594.17 m/sec) * 66.67 sec = 19286.84 meters
So, a reasonable solution would be to release the probe from a height of approximately 19,286.84 meters (about 19.29 km) and a distance of 2,000 meters from the landing location to ensure it lands with a speed no greater than 60 m/sec while the plane maintains a minimum horizontal speed of 30 m/sec.