To tackle this problem, we need to analyze the motion of the box as it interacts with the conveyor belt. The key here is to understand how the box's initial speed and the conveyor belt's speed affect its final speed when it comes to a stop on the belt. Let's break it down step by step.
Understanding the Motion
The box is initially moving at a speed of 2 m/s perpendicular to the conveyor belt, which is moving horizontally at 1 m/s. When the box is placed on the conveyor belt, friction will act to slow it down until it matches the speed of the belt.
Initial Velocities
- Speed of the box (initial): 2 m/s (perpendicular to the belt)
- Speed of the conveyor belt: 1 m/s (horizontal)
Velocity Components
We can break down the box's initial velocity into two components:
- Horizontal component (along the belt): 1 m/s (due to the belt)
- Vertical component (perpendicular to the belt): 2 m/s (initial speed of the box)
Calculating the Resultant Speed
When the box comes to a halt on the conveyor belt, it will eventually have the same horizontal speed as the belt, which is 1 m/s. However, we need to find the smallest speed of the box relative to the ground during this process.
Using the Pythagorean Theorem
To find the resultant speed of the box before it stops, we can use the Pythagorean theorem. The box's speed relative to the ground can be calculated as follows:
Resultant Speed (V) = √(V_horizontal² + V_vertical²)
Substituting the values:
V = √(1² + 2²) = √(1 + 4) = √5 ≈ 2.24 m/s
Final Thoughts
Thus, the smallest speed of the box with respect to the ground during this process, before it comes to a stop on the conveyor belt, is approximately 2.24 m/s. This speed accounts for both the horizontal motion of the conveyor and the initial vertical motion of the box. Understanding these components helps clarify how the box's motion changes as it interacts with the moving belt.
