Calculating the Horizontal Distance a Marble Falls from a Table

In this article, we will explore the physics behind the motion of a marble that rolls off a horizontal table and falls to the ground. We will apply the principles of free fall and horizontal velocity to determine the horizontal distance the marble covers as it falls. Let#8217;s break down the problem step by step and understand the important physical concepts involved.

Understanding the Problem

A question posed was about the horizontal distance from the edge of the table to the point where a marble strikes the ground. At first glance, the problem mentioned the table top speed and the time taken to strike the floor. However, there are some discrepancies in the stated details. If the marble were to fall freely from the edge of a table, the vertical component of the velocity after 1.5 seconds would be significant. For instance, the vertical distance traveled under free fall can be calculated using the formula:

Incorrect Assumption and Corrected Approach

The vertical velocity (v_y) after 1.5 seconds under free fall can be calculated using the equation:

[v_y g cdot t]

where (g approx 9.81 , text{m/s}^2) is the acceleration due to gravity. Substituting 1.5 seconds:

[v_y 9.81 , text{m/s}^2 times 1.5 , text{s} approx 14.715 , text{m/s}]

This should clarify that the given horizontal velocity of 5.6 m/s alone governs the horizontal motion, while the vertical motion follows the principles of free fall.

Calculating the Horizontal Distance

The key to solving the problem is understanding that the horizontal velocity remains constant during the fall. Given that the horizontal velocity is 5.6 m/s and the time of flight is 1.5 seconds, the horizontal distance (d) covered can be calculated using the formula:

[d v_x times t]

Substituting the given values:

[d 5.6 , text{m/s} times 1.5 , text{s} 8.4 , text{meters}]

This calculation clearly shows that if a marble leaves a horizontal table with a velocity of 5.6 meters per second, it will cover an additional 8.4 meters horizontally in 1.5 seconds under the influence of gravity.

Further Analysis: Horizontal Motion Over Time

For a more detailed analysis, let's break down the horizontal motion over time. In the first second, the marble will cover:

[5.6 , text{m/s} times 1 , text{s} 5.6 , text{meters}]

For the remaining 0.5 seconds, the distance covered will be:

[5.6 , text{m/s} times 0.5 , text{s} 2.8 , text{meters}]

Adding these distances together gives the total horizontal distance:

[5.6 , text{meters} 2.8 , text{meters} 8.4 , text{meters}]

This confirms our earlier calculation and provides insight into the horizontal acceleration vector over time in the projectile motion of the marble.

Conclusion

The horizontal distance a marble travels from the edge of a table to where it strikes the ground can be calculated using the principles of constant horizontal velocity and free fall vertical motion. Using a horizontal velocity of 5.6 meters per second and a flight time of 1.5 seconds, we determine the marble will travel a horizontal distance of 8.4 meters. This analysis highlights the importance of distinguishing between horizontal and vertical components in projectile motion.

Key Takeaways

The vertical component of velocity under free fall increases significantly in a short time period. The horizontal velocity remains constant during the fall. The total horizontal distance can be calculated by multiplying the horizontal velocity by the time of flight.