Question

A ball rolls off the top of a stairway with a horizontal velocity of magnitude \(5.0 \mathrm{ft} / \mathrm{s}\). The steps are \(8.0\) in. high and \(8.0\) in. wide. Which step will the ball hit first?

Short answer

To know the exact step, complete the steps in the solution. Calculate the time it takes for the ball to fall the height of one step, then how much horizontal distance it covers in this time, and finally calculate the step that the ball will hit. Ensure to have the distance in similar units before making computations, either all in feet or in inches.

Step-by-step solution

Compute Time it Takes for the Ball to Fall the Height of One Step

To calculate this, we consider that the vertical displacement is equal to the height of one step which is \(8.0\) in. We may need to convert this height into feet for consistency with the given horizontal velocity. One foot equals 12 inches, so a height of \(8.0\) in. can be written as \( \frac{8.0}{12} \) feet or \( \frac{2}{3} \) feet. The ball falls under the influence of gravity, therefore consider acceleration \(g = 32.2 \, \text{ft/s}^2\). Since initial vertical velocity is 0, use the equation of motion \(y = v_i t + \frac{1}{2} a t^2\) where \(y\) is the vertical displacement, \(v_i\) is the initial vertical velocity, \(t\) is time, and \(a\) is acceleration due to gravity. From this, \(t = \sqrt{\frac{2y}{g}}\).

Compute Horizontal Distance Traveled by the Ball

In projectile motion, the horizontal velocity remains constant. Therefore, the horizontal distance can be calculated by the product of time and the given horizontal velocity, \(v_h = 5.0 \, \text{ft/s}\). Use the equation of motion in a horizontal direction, \(x = v_h \cdot t\).

Determine Step the Ball will Hit

The problem becomes simple if the ball moves horizontally the same distance that it does vertically which means the ball will hit step 1. If the horizontal displacement is larger than the width of a step, then the ball will hit a following step. Find out the step that the ball will hit by dividing the horizontal displacement by the width of a step which is also \(8.0\) in or \( \frac{2}{3} \) ft. The result will give which step the ball will hit first.

Key concepts

Kinematics

Projectile motion is a fascinating concept within kinematics that deals with the movement of objects through the air. It importantly considers both horizontal and vertical motions simultaneously, although they are independent of each other. In projectile motion, the horizontal and vertical components are always treated separately due to differing influences.

• The horizontal motion is constant, as no external horizontal forces act on it (in an ideal situation), resulting in zero horizontal acceleration.
• The vertical motion, however, is influenced by gravity, causing the object to accelerate downward.

Understanding kinematics allows us to describe and predict the behavior of objects in motion, like a ball rolling off stairs.

Horizontal Velocity

In projectile motion, horizontal velocity is quite unique because it remains constant throughout the journey. This consistency arises because no horizontal forces, like air resistance (assuming it is negligible), act on the object. Therefore, the horizontal component of velocity does not change.
In the given problem, the ball starts with a horizontal velocity of 5.0 ft/s. As it rolls off the top of the stairs, this speed remains unaltered. It only moves in a straight line along the horizontal axis at this constant speed.

• Determine the horizontal distance by multiplying this constant velocity with time.
• For the ball, this means we need to consider the time it spends in transit before hitting the next step.

By keeping the horizontal motion constant, calculations on how far the ball travels before descending to the next step become more manageable.

Vertical Displacement

Vertical displacement refers to the distance the ball falls under gravity's influence. In this problem, the ball has no initial vertical velocity; it only begins to move vertically as gravity pulls it down after leaving the staircase.
Vertical displacement is critical for determining the actual path of the projectile and how it interacts with objects below it, like the steps in this problem.
To compute vertical displacement, we use the motion equation: \[ y = v_i t + \frac{1}{2} a t^2 \]where:

• \( y \) is the vertical displacement, or step height (\( \frac{2}{3} \) ft in this case).
• \( v_i \) is the initial vertical velocity, which is 0 here.
• \( a \) is the acceleration due to gravity \(32.2 \, \text{ft/s}^2\).

By solving for \( t \), we can find how long the ball remains airborne before hitting the next step. This time is used later for horizontal displacement calculations.

Acceleration due to Gravity

Gravity plays a fundamental role in projectile motion, specifically influencing the vertical component of an object's trajectory. The acceleration due to gravity is a constant denoted by \( g \), which measures \( 32.2 \, \text{ft/s}^2 \) for this exercise.
Gravity is what causes the ball to accelerate downwards once it rolls off the stairs. The time it takes to descend is crucial because it helps determine if the ball will hit a step or travel further.

• The acceleration formula we use is part of the equations of motion for vertical displacement.
• Since the initial vertical speed of the ball is zero, the only force acting on it is gravity.
• This creates the downward acceleration, resulting in a parabolic trajectory typical in projectile motion.

Understanding gravity's impact helps predict where the ball will land, as it directly influences how quickly it falls, impacting the overall outcome of the exercise.