To analyze the situation where two identical balls are interconnected by a massless and inextensible thread in a gravity-free environment, we need to consider their motions and how they interact with each other. Given that one ball moves directly towards the other with a velocity \( v \), while the second ball moves perpendicularly with the same velocity \( v \), we can determine when the thread becomes taut.
Understanding the Motion of the Balls
Let's denote the two balls as Ball A and Ball B. At time \( t = 0 \):
- Ball A is moving towards Ball B along the x-axis.
- Ball B is moving perpendicular to Ball A along the y-axis.
Since both balls are identical and start from the same point, we can set their initial positions as follows:
- Position of Ball A: \( (0, 0) \)
- Position of Ball B: \( (0, 0) \)
Calculating the Positions Over Time
As time progresses, the positions of the balls can be expressed as:
- Position of Ball A at time \( t \): \( (vt, 0) \)
- Position of Ball B at time \( t \): \( (0, vt) \)
Now, we need to find the distance between the two balls as a function of time. The distance \( d \) between the two balls at any time \( t \) can be calculated using the Pythagorean theorem:
Distance Calculation
The distance \( d \) between Ball A and Ball B is given by:
\( d(t) = \sqrt{(vt)^2 + (vt)^2} = \sqrt{2(vt)^2} = vt\sqrt{2} \)
Determining When the Thread Becomes Taut
The thread becomes taut when the distance between the two balls equals the length of the thread. Since the thread is just taut at the beginning, we can assume its length is zero at \( t = 0 \). However, as the balls move, the distance between them increases until the thread is fully extended.
To find the time \( t \) when the thread becomes taut, we need to consider the geometry of the situation. The thread will become taut when the distance \( d(t) \) equals the initial separation distance, which is zero. However, since the balls are moving towards each other, we can set the distance equal to the length of the thread, which we can denote as \( L \). For this case, we can assume \( L \) is the distance at which the thread becomes taut:
Setting Up the Equation
We can express this as:
\( vt\sqrt{2} = L \)
Solving for \( t \), we have:
\( t = \frac{L}{v\sqrt{2}}
\)
Conclusion
In summary, the time at which the thread becomes taut is given by the equation \( t = \frac{L}{v\sqrt{2}} \). This shows that the time depends on the initial length of the thread and the velocities of the balls. If the length of the thread is zero, the balls will never become taut, but if it has a finite length, this equation will give you the exact time at which the thread becomes taut as the balls move towards each other.
