An airplane moving horizontally at 150 m/s drops a package from an altitude is 490 m. How long does it take the package to reach the ground? How far, horizontally, from the spot over which it was dropped does the package land?
Well, well, well, we've got a gravity-defying package on our hands! Let me calculate this comical situation for you.
To find out how long it takes for the package to reach the ground, we can use the good old formula for time, which is: time = distance / speed.
Now, since the package is dropping vertically, we don't need to worry about the horizontal speed of the airplane. We just need to focus on the package's free fall. The distance it falls is 490 m, and its speed (or should I say lack of speed) is due to gravity, not the airplane.
So, using the formula, time = 490 m / 9.8 m/s² (because that's the acceleration due to gravity), we find that the package takes approximately 50 seconds to reach the ground.
Now, let's talk about where this mischievous package lands horizontally. Since the airplane is moving at a constant horizontal speed of 150 m/s, the package is also carried forward during its free fall.
To find how far the package lands horizontally from the drop point, we multiply the time (50 seconds) by the horizontal speed (150 m/s). That gives us a grand total of 7,500 meters!
So, congratulations! Your package, with its amazing parachute-free descent, lands approximately 7.5 kilometers away from the spot it was dropped. I hope it doesn't scare any unsuspecting villagers along the way!
To find the time it takes for the package to reach the ground, we can use the equation of motion:
\[ h = \frac{1}{2}gt^2 \]
where \( h \) is the initial drop height and \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)). Rearranging the equation to solve for time:
\[ t = \sqrt{\frac{2h}{g}} \]
Substituting the given values:
\[ t = \sqrt{\frac{2 \times 490}{9.8}} \]
\[ t = \sqrt{100} \]
\[ t = 10 \, \text{s} \]
So, it takes the package 10 seconds to reach the ground.
To find how far, horizontally, the package lands from the spot over which it was dropped, we can use the equation:
\[ d = v \cdot t \]
where \( d \) is the horizontal distance traveled, \( v \) is the horizontal velocity of the airplane (150 m/s), and \( t \) is the time of flight (10 s). Substituting the given values:
\[ d = 150 \cdot 10 \]
\[ d = 1500 \, \text{m} \]
So, the package lands 1500 meters horizontally from the spot over which it was dropped.
To find the time it takes for the package to reach the ground, we can use the formula for the time of flight in free fall. The package is dropped from rest, so the initial vertical velocity is 0 m/s. The acceleration due to gravity is approximately 9.8 m/s^2.
Using the formula:
time = sqrt(2 * vertical_distance / acceleration due to gravity)
Given that the vertical distance is 490 m and the acceleration due to gravity is 9.8 m/s^2, we can calculate the time it takes for the package to reach the ground:
time = sqrt(2 * 490 m / 9.8 m/s^2) = 10 s.
Therefore, it takes 10 seconds for the package to reach the ground.
To find how far, horizontally, from the spot over which it was dropped the package lands, we can use the formula for horizontal distance traveled:
horizontal_distance = horizontal_velocity * time
Given that the horizontal velocity is 150 m/s and the time is 10 s, we can calculate the horizontal distance:
horizontal_distance = 150 m/s * 10 s = 1500 m.
Therefore, the package lands 1500 meters horizontally from the spot over which it was dropped.
time to fall: 9.81 t^2 = 490
horizontal distance is 150t