Stones are thrown horizontally with the same velocity from the tops of two different buildings. One stone lands twice as far from the base of the building from which it was thrown as does the other stone. Find the ratio of the height of the taller building to the height of the shorter building.
The fall time of the stone that travels twice as far is twice as long.
Building height is proportional to the SQUARE of the fall time.
The taller building is therefore four times taller than the short one.
To solve this problem, let's first analyze the motion of the stones.
Since the stones are thrown horizontally with the same initial velocity, we can disregard the effects of gravity in the horizontal direction. This means that both stones will have the same horizontal displacement and time of flight.
Let's assume the heights of the shorter and taller buildings are h1 and h2, respectively. We need to find the ratio h2/h1.
The horizontal displacement of the stone thrown from the shorter building is equal to twice the distance it lands from the base. Let's call this distance "S1." So, the horizontal displacement for the stone thrown from the shorter building is 2S1.
The horizontal displacement of the stone thrown from the taller building is equal to the distance it lands from the base. Let's call this distance "S2." So, the horizontal displacement for the stone thrown from the taller building is S2.
Since both stones have the same initial horizontal velocity and time of flight, we can use the equation of motion in the horizontal direction:
S = Vx * t
Where S is the horizontal displacement, Vx is the horizontal velocity, and t is the time of flight.
For both stones, since they have the same initial horizontal velocity, we can write:
2S1 = S2
Now, let's analyze the motion vertically.
The time of flight for both stones can be calculated using the equation:
t = sqrt(2h/g)
Where h is the height of the building and g is the acceleration due to gravity.
For the stone thrown from the shorter building, the time of flight is:
t1 = sqrt(2h1/g)
For the stone thrown from the taller building, the time of flight is:
t2 = sqrt(2h2/g)
Since both stones have the same time of flight, we have:
t1 = t2
Substituting the expressions for t1 and t2, we get:
sqrt(2h1/g) = sqrt(2h2/g)
Squaring both sides, we have:
2h1/g = 2h2/g
Simplifying, we find:
h1 = h2
Therefore, the ratio of the height of the taller building (h2) to the height of the shorter building (h1) is 1:1, meaning they have the same height.