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Initial velocity, denoted as \( v_0 \), is essentially the speed at which an object starts moving. It is a pivotal concept in linear motion, as it sets the stage for how the object will continue to move over time under certain conditions. In the context of the given problem, the initial velocity is the speed of the ball at the very start, as it begins to roll down the inclined plane.
The initial velocity serves as the first term in the velocity-time equation, which ultimately helps to predict the ball's speed at different moments. Understanding initial velocity is crucial because:

• It provides a baseline from which any calculations of future velocity can be considered.
• It influences the overall motion of an object as it combines with other forces like acceleration.

When determining initial velocity from experimental data, like in our problem where \( v(2) = 16 \) and \( v(5) = 25 \), you can set up equations based on these known velocities. Solving them enables you to calculate \( v_0 \) alongside acceleration using given conditions. In our scenario, after solving the system of equations, we find that \( v_0 = 10 \ \text{cm/sec} \). This signifies that the initial speed of the ball was 10 cm/sec before any additional acceleration affected it.

Acceleration is the rate at which an object's velocity changes over time. In the context of linear motion, it's a constant factor that either speeds up or slows down the object in motion. In our problem, the acceleration \( a \) is given in \( \text{cm/sec}^2 \), which indicates how much the velocity changes per second.
For the ball rolling down the inclined plane, acceleration acts as a driving force that increases its velocity as time progresses, assuming there is no other resisting force.

Some key aspects of acceleration include:

• It can be positive (increasing velocity) or negative (decreasing velocity, often called deceleration).
• An object with zero acceleration is moving at a constant velocity.

To find the acceleration in our problem, we utilized the equation \( v(t) = v_0 + at \). By setting up the system of equations from known values \( v(2) = 16 \) and \( v(5) = 25 \), and solving them, we determined the acceleration to be \( a = 3 \ \text{cm/sec}^2 \). This means that every second, the ball's velocity increases by 3 cm/sec.

The velocity-time equation summarizes how the velocity of an object evolves with time under linear motion conditions. This formula, expressed as \( v(t) = v_0 + at \), connects initial velocity, acceleration, and time, making it a powerful tool for predicting future movements.
This equation allows us to calculate the velocity at any given moment if we know the initial velocity and the acceleration. In other words, it provides a comprehensive view of how the object's speed changes over time. The terms in the equation are:

• \( v(t) \): The velocity at time \( t \).
• \( v_0 \): The initial velocity, or the speed at which the object commenced moving.
• \( at \): The term representing the change in velocity due to constant acceleration over time.

By substituting different time values, we can ascertain the velocity at those specific moments. In our exercise, knowing \( v(2) = 16 \) and \( v(5) = 25 \), we plugged these into the velocity-time equation to produce two equations:\
- \( v_0 + 2a = 16 \)
- \( v_0 + 5a = 25 \)
Solving them helped in finding both \( v_0 \) and \( a \). This demonstrates the utility of the velocity-time equation in planning and analyzing motion.