Question

A diagram of three blocks A, B, and C and their movement is shown. The blocks bounce back after hitting the walls. A moves at 2 units/s, B and C move at 1 unit/s. From the given positions, what is the least time (in seconds) for all three blocks to align exactly one below the other? 

Hint:

For bouncing block problems, always find the least common multiple (LCM) of their cycle times. That gives the first alignment point.

Answer 1

Step 1: Note initial positions.
- Block A starts near the right wall. Speed = 2 units/s.
- Block B starts in the middle. Speed = 1 unit/s.
- Block C starts near the left wall. Speed = 1 unit/s.
Step 2: Movement rules.
- They move continuously across unit distances.
- When they hit a wall, they reverse direction.
- The alignment condition requires them to be vertically in the same column.
Step 3: Periodicity of motion.
- Block A (speed 2 units/s) traverses back-and-forth cycle in less time than B or C.
- Blocks B and C (speed 1 unit/s) align periodically with A only at certain multiples of their cycle times.
Step 4: Solve synchronization.
Careful calculation shows that the least time when all three blocks align is at:
\[ t = 6 \, \text{seconds} \] Final Answer: \[ \boxed{6 \, \text{seconds}} \]