91Ó°ÊÓ

Kinematic equations are fundamental to the study of motion. They describe the movement of objects through various types of motion—linear, projectile, or rotational—by relating the parameters: velocity, acceleration, time, and displacement. These equations assume constant acceleration, making them perfect for analyzing free-fall and projectile scenarios like our staircase problem.
In the case of projectile motion, we often break down movements into vertical and horizontal components. The vertical motion is influenced by the constant gravitational acceleration, while the horizontal motion occurs at a uniform velocity without acceleration.
For vertical motion, we frequently use the equation:

• \( y = rac{1}{2}gt^2 \) where \( y \) is the height, \( g = 9.8 \,\text{m/s}^2 \) is the acceleration due to gravity, and \( t \) is the time.

For horizontal motion, when dealing with a projectile like a rolling ball, we use:

• \( x = v_x t \) where \( x \) is the horizontal distance traveled, and \( v_x \) is the horizontal velocity.

These two equations help break the problem into manageable calculations, allowing us to find the time it takes to fall and the necessary horizontal speed.

Vertical motion is an essential aspect of projectile motion. It refers to how objects move under the influence of gravity, without horizontal forces affecting their path.
In the context of our problem, the ball is subject to a free fall from the top of the staircase to the lowest step, a height of \(0.3\,\text{m}\). Gravity acts downward at \(9.8\,\text{m/s}^2\), and this force changes the ball's vertical position over time.
The equation \(y = \frac{1}{2}gt^2\) helps calculate the time (\(t\)) it takes for the ball to reach the bottom. Here, \(y\) is the initial height (0.3 m) and substituting it in the equation allows finding how long it takes to fall through this height.

• By rearranging, \( t^2 = \frac{0.3 \times 2}{9.8} \).
• Solve for \(t\) to get approximately \(0.247\,\text{s}\).

This value, \(t\), becomes pivotal as it is used to find the required horizontal velocity to ensure the ball lands directly on the lowest step.

Understanding horizontal motion is crucial for solving the problem of a ball rolling off a staircase. Here, the concern is the ball's travel parallel to the ground without vertical acceleration.

While free-falling vertically, the ball must simultaneously travel a horizontal distance equal to the combined width of two steps, totaling \(0.4\,\text{m}\). Unlike vertical motion, horizontal motion in projectile problems typically involves constant velocity since no horizontal force acts on the ball (assuming minimal air resistance).

The equation used here is \(x = v_x \cdot t\), where \(x\) is the horizontal distance (0.4 m) and \(v_x\) is the unknown horizontal velocity. Using the pre-determined time of vertical descent, \(t = 0.247\,\text{s}\), we can substitute in the equation:

• \( 0.4 = v_x \cdot 0.247 \), where solving for \(v_x\) gives approximately \(1.62\,\text{m/s}\).

This velocity ensures that while the ball falls vertically, it simultaneously covers enough distance to land on the intended lower step.