Certainly! Let's break down the question and understand the scenario step by step.
### Problem Breakdown:
1. We have a ball rolling down a flat, frictionless ramp.
2. The ball is rolling with a constant velocity of 13 m/s.
3. We need to determine the acceleration of the ball over two different durations: three seconds and over an infinite number of seconds.
### Key Concepts:
1. Constant Velocity: When an object moves with a constant velocity, it means that its speed and direction of motion do not change.
2. Acceleration: Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, it's given by:
[tex]\[
\text{Acceleration} (a) = \frac{\text{Change in Velocity} (\Delta v)}{\text{Change in Time} (\Delta t)}
\][/tex]
3. Frictionless Surface: A frictionless surface implies that there are no resistive forces acting on the ball. Hence, the ball will not speed up or slow down due to friction.
### Solution:
#### Over Three Seconds:
1. Initial velocity (u): 13 m/s
2. Final velocity (v): Since the velocity is constant, the final velocity remains 13 m/s.
3. Time interval (t): 3 seconds.
Since there is no change in velocity (u = v):
[tex]\[
\Delta v = v - u = 13\, \text{m/s} - 13\, \text{m/s} = 0\, \text{m/s}
\][/tex]
Using the formula for acceleration:
[tex]\[
a = \frac{\Delta v}{\Delta t} = \frac{0\, \text{m/s}}{3\, \text{seconds}} = 0\, \text{m/s}^2
\][/tex]
Hence, the acceleration of the ball over three seconds is [tex]\(0 \,\text{m/s}^2\)[/tex].
#### Over an Infinite Number of Seconds:
1. Initial velocity (u): 13 m/s
2. Final velocity (v): Again, since the velocity is constant, the velocity remains 13 m/s over any period.
3. Time interval (t): Infinite.
Similarly, with no change in velocity:
[tex]\[
\Delta v = v - u = 13\, \text{m/s} - 13\, \text{m/s} = 0\, \text{m/s}
\][/tex]
Using the formula for acceleration:
[tex]\[
a = \frac{\Delta v}{\Delta t} = \frac{0\, \text{m/s}}{\infty} = 0\, \text{m/s}^2
\][/tex]
Therefore, the acceleration of the ball over an infinite number of seconds is [tex]\(0 \,\text{m/s}^2\)[/tex].
### Explanation:
The ball is rolling at a constant velocity, which means there's no change in its speed. Acceleration only occurs when there's a change in velocity—either speeding up, slowing down, or changing direction. Since neither of these is happening in the given scenario, the ball's acceleration remains zero regardless of how much time has passed.
In summary:
- The acceleration of the ball over three seconds is [tex]\(0 \,\text{m/s}^2\)[/tex].
- The acceleration of the ball over an infinite number of seconds is also [tex]\(0 \,\text{m/s}^2\)[/tex].
