Math Problem Statement
Solution
The problem you've provided involves calculating the shortest and longest distances, , where the end of the cart can be placed so that the packages fall inside the cart. Here's a breakdown of how to solve it:
Given Data:
- The conveyor belt speed:
- Angle of the conveyor belt:
- Height of drop from conveyor belt to cart:
- Length of the cart:
We need to determine the range of values for (distance from the conveyor belt to the cart) where the packages fall into the cart, i.e., the shortest distance and longest distance .
Step 1: Vertical Motion
The time it takes for the package to fall from the conveyor belt to the cart is determined by the vertical motion. Using the equation of motion for free fall:
Solving for , we get:
Step 2: Horizontal Motion
The horizontal velocity of the packages is given by the component of the conveyor belt speed along the horizontal direction. Since the belt is inclined at , the horizontal velocity is:
The horizontal distance the package travels is:
Step 3: Determine Range for
The package must fall within the length of the cart, which is 1 meter. So, the distance where the cart can be placed must be such that the package falls between and .
- The shortest distance is when the package lands at point :
- The longest distance is when the package lands at point , which is 1 meter further than :
Final Answer:
The range of distances where the cart can be placed is:
Would you like further details on any of the steps?
Related Questions:
- How does the time of fall change with different heights?
- What happens if the conveyor belt speed is increased to 3 m/s?
- How would the calculation differ if the incline angle were 45 degrees?
- What is the horizontal distance if the cart is moved 0.5 meters closer?
- How would the range of be affected if the cart length were increased to 1.5 meters?
Tip:
When dealing with inclined planes, always break down velocities into their horizontal and vertical components to simplify the problem.
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Math Problem Analysis
Mathematical Concepts
Projectile Motion
Free Fall
Trigonometry
Formulas
Vertical motion equation: h = (1/2) * g * t^2
Horizontal motion equation: d = v_x * t
Horizontal velocity: v_x = v_c * cos(θ)
Theorems
Equations of motion for free fall
Trigonometric decomposition of velocity
Suitable Grade Level
Grades 10-12
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