Stones are thrown horizontally with the same velocity from two buildings. One stone lands 1.81 as far away from its building as the other stone. Determine the ratio of the heights of the two buildings.
time 2 = 1.81 * time 1
h = (1/2)gt^2
so
h/t^2 = constant = H1/t1^2 = H2/t2^2
H2/H1 = t2^2/t1^2 = 1.81^2 = 3.27
To determine the ratio of the heights of the two buildings, we can use the principles of projectile motion.
Let's denote the height of the first building as h1 and the height of the second building as h2. To simplify the problem, we assume that the stones are thrown from the ground level.
When an object is thrown horizontally, it follows a parabolic trajectory. The horizontal displacement, or range (R), of a projectile can be given by the equation:
R = v * t
where v is the horizontal velocity of the projectile and t is the time of flight. Since the stones are thrown horizontally with the same velocity, their ranges will be proportional to their time of flight.
Now, let's consider the first stone thrown from the first building. Its range is 1.81 times the range of the second stone. Mathematically, we can express this as:
R1 = 1.81 * R2
Since both stones are thrown horizontally, their horizontal velocities are identical. We can cancel out the v terms from the equation, resulting in:
t1 = 1.81 * t2
Next, we need to find the expression for the time of flight (t) for each stone. In projectile motion, the vertical distance traveled (h) can be related to the time of flight using the equation:
h = (1/2) * g * t^2
where g is the acceleration due to gravity. Since both stones are thrown from the ground level, the initial vertical position for both is zero.
For the first stone thrown from the first building, its vertical distance traveled can be expressed as:
h1 = (1/2) * g * t1^2
For the second stone thrown from the second building, its vertical distance traveled can be expressed as:
h2 = (1/2) * g * t2^2
Using the expression for the time ratio (t1 = 1.81 * t2), we can substitute it into the height equations:
h1 = (1/2) * g * (1.81 * t2)^2
h2 = (1/2) * g * t2^2
Now, we can divide the equation for h1 by h2 to find the ratio of the heights of the buildings:
(h1 / h2) = [(1/2) * g * (1.81 * t2)^2] / [(1/2) * g * t2^2]
We can simplify the equation further:
(h1 / h2) = (1.81^2 * t2^2) / (t2^2)
(h1 / h2) = 1.81^2
(h1 / h2) = 3.2761
Therefore, the ratio of the heights of the two buildings is approximately 3.2761.