Stones are thrown horizontally with the same velocity from the tops of two different buildings. One stone lands four times 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.
See previous post: 9-16-13, 8:40 PM.
To solve this problem, let's break it down step-by-step:
Step 1: Assign variables
Let's assign variables to the given information. Let:
- h1 = height of the first building
- h2 = height of the second building
- d1 = distance traveled by the stone thrown from the first building
- d2 = distance traveled by the stone thrown from the second building
Step 2: Define the information given
According to the problem, one stone lands four times as far from the base of the building from which it was thrown as does the other stone. This can be expressed as:
d1 = 4 * d2
Step 3: Use kinematic equations
Since the stones are thrown horizontally, we can use the kinematic equation for horizontal motion:
d = v * t
where d is the distance, v is the initial velocity, and t is the time.
Since the stones are thrown horizontally with the same velocity, we can equate their times of flight. The time of flight can be expressed as:
t = d / v
Step 4: Solve for time of flight
For the stone thrown from the first building:
t1 = d1 / v
For the stone thrown from the second building:
t2 = d2 / v
Since the times of flight are equal, we can equate the equations:
d1 / v = d2 / v
d1 = d2
Step 5: Solve for the ratio of building heights
Since both stones cover the same horizontal distance, the ratio of their respective heights is the same as the ratio of their horizontal distances:
h1 / h2 = d1 / d2
But we know that d1 = 4 * d2
Therefore:
h1 / h2 = 4 * d2 / d2
h1 / h2 = 4
So, the ratio of the height of the taller building (h1) to the height of the shorter building (h2) is 4:1.