A marble rolls off a table with a horizontal velocity of 1.2 m/s. The marble falls in a cup placed horizontally 0.51m from the table's edge. How high is the table?
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To determine the height of the table, we need to analyze the motion of the marble after it rolls off the table and before it falls into the cup.
First, let's break down the motion of the marble horizontally and vertically.
1. Horizontal Motion:
The marble has a horizontal velocity of 1.2 m/s, meaning it maintains a constant speed in the horizontal direction. Since there are no external forces acting on the marble in the horizontal direction, its horizontal velocity remains the same throughout its motion.
2. Vertical Motion:
The marble is subject to the force of gravity, causing it to accelerate vertically downward. We can use the equations of motion to analyze its vertical motion.
Let's consider the vertical displacement of the marble. We can use the formula for vertical displacement with respect to time:
y = vt + (1/2)gt^2,
where y is the vertical displacement, v is the initial vertical velocity, t is the time, and g is the acceleration due to gravity.
In this scenario, at the moment the marble rolls off the table, its initial vertical velocity is 0 because it only has a horizontal velocity. Therefore, the equation simplifies to:
y = (1/2)gt^2.
The horizontal distance traveled by the marble before falling into the cup is given as 0.51 m. To find the corresponding time it takes for the marble to reach the cup horizontally, we can use the formula:
d = vt,
where d is the horizontal distance, v is the horizontal velocity, and t is the time.
Solving for t:
t = d / v = 0.51 m / 1.2 m/s.
Now we can substitute this time into the equation for vertical displacement:
y = (1/2)g(0.51 m / 1.2 m/s)^2.
Using the acceleration due to gravity, which is approximately 9.8 m/s^2, we can solve for y to find the height of the table.