91Ó°ÊÓ

When we talk about uniform acceleration in physics, we're referring to a situation where an object speeds up or slows down at a constant rate. This means the acceleration does not change over time. In our exercise, the ball rolls down a hill under the influence of gravity, experiencing uniform acceleration throughout its motion. This makes things a bit simpler because the equations we use to describe the motion can take advantage of this constancy.
Uniform acceleration is a key concept when handling problems where the speed of an object increases or decreases consistently. It allows us to predict how far and fast an object will travel over a specific period without needing to know all the intermediate values of speed. Such situations are common in things like a car accelerating from a stoplight or, in this case, a ball rolling down a hill. The predictability here is useful for solving the exercise, where knowing the acceleration lets us find out how far the ball has rolled at various times.

Equations of motion are a set of formulas that describe the mathematical relationship between displacement, velocity, acceleration, and time. In kinematics, these equations are crucial in solving problems involving objects in motion. They allow us to find unknown values such as final velocity or distance traveled, given certain variables.
For our rolling ball, we use the equation \( s = ut + \frac{1}{2}at^2 \), which lets us find the distance covered when we know the initial velocity \( u \), the acceleration \( a \), and the time \( t \). Because the ball starts from rest, \( u = 0 \), simplifying our task. Thus, we only need to focus on the acceleration part of the equation. Using these equations, we effectively calculated the distance traveled by the ball during both the first and second intervals.

Initial velocity is a key element in physics problems, describing the speed and direction of an object when it first starts moving. It forms the basis of what happens in any motion thereafter. In our problem, since the ball is starting from rest, its initial velocity \( u \) is 0. This simplifies the calculations significantly because it removes the need to account for any initial speed in the motion equations.
By knowing the initial velocity, we can apply it to determine how acceleration affects the subsequent motion. This makes calculations more straightforward when using the equations of motion. When an object starts from rest, just like the ball rolling down the hill, focusing on how the acceleration contributes to the distance traveled becomes the primary task.