Problem 50 - College Physics

Question

Ball A rolls at a constant speed of \(0.25 \mathrm{~m} / \mathrm{s}\) on a table \(0.95 \mathrm{~m}\) above the floor, and ball \(\mathrm{B}\) rolls on the floor directly under the first ball with the same speed and direction. (a) When ball A rolls off the table and hits the floor, (1) ball B is ahead of ball A, (2) ball B collides with ball \(A,(3)\) ball \(A\) is ahead of ball \(B\). Why? (b) When ball A hits the floor, how far from the point directly below the edge of the table will each ball be?

Step-by-Step Solution

Answer

When Ball A hits the floor, both balls are 0.11 m from the table edge and aligned horizontally.

Step 1: Analyze Ball A's Motion

Ball A rolls off a table and falls due to gravity. It initially has a horizontal velocity of \(0.25\, \text{m/s}\) and experiences an acceleration of \(9.81\, \text{m/s}^2\) downward. We need to determine the time it takes for Ball A to fall \(0.95\, \text{m}\) to the floor.

Step 2: Calculate Time for Ball A to Hit the Floor

Use the formula for vertical motion: \(s = ut + \frac{1}{2}gt^2\), where \(s = 0.95\, \text{m}, u = 0\, \text{m/s}\) (as there is no initial vertical velocity), and \(g = 9.81\, \text{m/s}^2\). Solve for \(t\): \[0.95 = \frac{1}{2} \times 9.81 \times t^2\] Solving gives \(t \approx 0.44\, \text{seconds}\).

Step 3: Determine Horizontal Distance Traveled by Ball A

With a horizontal velocity of \(0.25\, \text{m/s}\), the horizontal distance \(d_A\) Ball A travels in \(0.44\, \text{seconds}\) is:\[d_A = 0.25 \times 0.44 = 0.11\, \text{m}\]

Step 4: Analyze Ball B's Motion

Ball B rolls on the floor at the same speed and in the same direction as Ball A but does not fall. Thus, it travels horizontally for the same duration, \(0.44\, \text{seconds}\).

Step 5: Calculate Distance Traveled by Ball B

Since Ball B also has a speed of \(0.25\, \text{m/s}\) and rolls for \(0.44\, \text{seconds}\), the distance \(d_B\) it travels is:\[d_B = 0.25 \times 0.44 = 0.11\, \text{m}\]

Step 6: Compare Positions of Ball A and Ball B when Ball A Hits the Floor

Both balls roll at the same horizontal speed and for the same time, so both travel \(0.11\, \text{m}\) horizontally. Thus, when Ball A hits the floor, Ball B is neither ahead nor behind Ball A; they are in line with each other.

Key Concepts

Horizontal Motion

Projectile motion involves movements in both horizontal and vertical directions. Horizontal motion, in particular, is the motion parallel to the ground. In our problem, Ball A initially moves horizontally with a constant speed of \(0.25 \, \text{m/s}\).
It retains this speed as there are no horizontal forces acting on it, meaning there's no horizontal acceleration. The same applies to Ball B, which rolls on the floor directly below where Ball A starts. Since both balls experience no horizontal forces, we can calculate the horizontal distance they travel using the formula:

• Distance = Speed × Time

Understanding the horizontal motion helps us analyze the movement of objects along the ground or any horizontal plane as they cover equal distances over equal periods. This concept is key in analyzing how far projectile-like motions travel horizontally when no horizontal forces are involved.

Vertical Motion

Vertical motion is an essential part of projectile movement. It refers to motion along the vertical direction, affected by gravity. In the exercise, only Ball A experiences vertical motion because it rolls off the table and falls.
Gravity acts as the vertical force here, accelerating Ball A downwards at \(9.81 \, \text{m/s}^2\). Initially, Ball A has no vertical speed as it rolls off horizontally from the table top. Its vertical distance covered can be calculated using::

• \(s = ut + \frac{1}{2}gt^2\)

Here, \(s\) is the vertical distance fallen (0.95 m), \(u\) the initial vertical speed (0), \(g\) the acceleration due to gravity, and \(t\) the time it takes to fall. Understanding vertical motion is crucial, as it dictates the time a projectile remains in the air.

Uniform Acceleration

Uniform acceleration occurs when an object's velocity changes at a constant rate. In our scenario, Ball A undergoes uniform acceleration due to gravity. Since it falls off a table, its speed increases steadily as it descends.
This acceleration is crucial in calculating how quickly and for how long Ball A falls. The familiar formula \(s = ut + \frac{1}{2}gt^2\) helps us link the variables of initial speed, time, distance, and acceleration together. It's vital to note:

• Uniform acceleration doesn't affect horizontal speed if no horizontal forces exist.
• Gravity causes constant acceleration downward at \(9.81 \, \text{m/s}^2\).

Understanding uniform acceleration provides insights into how moving objects like projectiles develop speed over time, particularly vertically in free fall scenarios.

Kinematics

Kinematics is the study of motion without considering the forces causing it. In this exercise, kinematics helps analyze Ball A's and Ball B's movements.
Both balls have the same initial horizontal speed, but only Ball A undergoes vertical velocity change due to gravity. Kinematic equations assist in calculating:

• Time taken for Ball A to reach the ground.
• Horizontal distances traveled by both balls.

Kinematics relies on knowing initial speeds, distance, time, and acceleration to predict an object's position at any given moment. By using kinematics, we can understand projectile paths and compare them, as when assessing Ball A and Ball B, confirming they align horizontally upon Ball A's landing.