To understand the problem involving the three stones thrown from a tower, we need to analyze the motion of each stone using the principles of kinematics. The key here is to recognize how the initial conditions (like the direction and speed of the throw) affect the time it takes for each stone to reach the ground. Let's break down the situation step by step.
Understanding the Motion of Each Stone
We have three stones, each released under different conditions:
- Stone 1: Thrown vertically upwards with an initial speed.
- Stone 2: Thrown vertically downwards with the same initial speed.
- Stone 3: Released from rest.
Analyzing the First Stone (Thrown Upwards)
When the first stone is thrown upwards, it will first ascend until it reaches its maximum height, then it will fall back down to the ground. The time taken for this entire journey is denoted as t1. The equations of motion tell us that the total time can be expressed in terms of the initial velocity and the height of the tower.
Examining the Second Stone (Thrown Downwards)
The second stone, thrown downwards with the same initial speed, will reach the ground in time t2. Since it is thrown downwards, it will take less time to reach the ground compared to the first stone. The kinematic equations again apply here, and we can express t2 in terms of the height and the initial speed.
Considering the Third Stone (Released from Rest)
The third stone is simply released from rest, meaning it starts with zero initial velocity. The time taken for this stone to reach the ground is t3. This time is typically longer than t1 and shorter than t2, depending on the height of the tower.
Deriving the Relationships
Now, let's derive the relationships given in the problem:
- Equation 1: t1 + t2/2 = t3
- Equation 2: 1/t3 = 1/t2 - 1/t1
- Equation 3: t3^2 = t1^2 - t2^2
These equations can be derived from the kinematic equations for uniformly accelerated motion. For instance, the first equation can be interpreted as a balance of the times taken by the stones, while the second equation relates the reciprocal of the times, which is a common technique in physics to relate different rates of motion.
Verifying the Answer
To verify that the answer is indeed B, we can substitute the values of t1, t2, and t3 into the equations and check for consistency. The relationships derived from the kinematic equations should hold true, confirming that the time relationships are valid.
In summary, the problem illustrates the principles of motion under gravity and how initial conditions affect the time taken for objects to fall. By analyzing each stone's motion and deriving the relationships, we can confidently arrive at the conclusion that the answer is B.
