To determine the height at which you need to throw an object so that it falls and lands at exactly the same time as if you dropped it from your hand, we need to consider the concept of "hang time" or the time it takes for an object to reach the ground when thrown horizontally.
When you throw an object horizontally, it follows a parabolic trajectory, while when you drop an object, it falls straight down. For the two motions to have the same time of descent, the horizontal distance covered by the thrown object during its hang time should be the same as the vertical distance covered by the dropped object during the same time.
The time it takes for an object to fall vertically from rest can be calculated using the equation for free fall:
t = sqrt((2h)/g),
where: t is the time of descent, h is the vertical distance or height, g is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
To find the horizontal distance traveled by the thrown object during the same time, we use the equation:
d = v*t,
where: d is the horizontal distance, v is the initial horizontal velocity.
Since the initial horizontal velocity is constant, we can set the vertical distance h and the horizontal distance d equal to each other:
h = v*t.
Substituting the value of t from the equation for free fall:
h = v*sqrt((2h)/g).
Now, we can solve this equation to find the height h. By squaring both sides of the equation, we get:
h² = (v²/g)*(2h).
Simplifying further:
h² = (2v²/g)*h.
Dividing both sides of the equation by h:
h = 2v²/g.
Therefore, the height at which you need to throw an object for it to fall and land at the same time as if you dropped it from your hand is given by:
h = (2v²/g),
where v is the initial horizontal velocity.
Keep in mind that this equation assumes there is no air resistance, and the object is thrown horizontally without any vertical component to its initial velocity.