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Kinematics involves the study of motion without considering the forces that cause it. In the context of our hockey puck, we analyze its position, velocity, and acceleration over time. In the initial scenario, the puck's velocity at time \( t = 0 \) is 3.00 m/s to the right. This starting point gives us valuable data for predicting how the puck's velocity changes when forces are applied.

To understand these changes, we rely on equations that connect velocity, acceleration, and time. Specifically, the equation \( v_f = v_i + at \) helps us predict the puck's final velocity. Here, \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is acceleration, and \( t \) is time.

By breaking down motion into these components, kinematics provides a clear picture of the puck’s trajectory. This baseline understanding is essential when adding the complexities of force and acceleration.

Newton’s Second Law tells us that force equals mass times acceleration, or \( F = ma \). This law is key to understanding how motion changes when forces act on objects like our hockey puck.

When a force is applied, it changes the puck’s velocity by creating acceleration. In our exercise with the puck, two different scenarios reveal how this works:

• When a force of 25.0 N pushes to the right, it results in a calculated acceleration of \( 156.25 \text{ m/s}^2 \).
• When a force, instead, of 12.0 N acts to the left, the acceleration becomes \( -75.0 \text{ m/s}^2 \), indicating reversed motion due to the opposite direction of force.

Regardless of whether the force moves the puck faster or slows it down, understanding the relation between force, mass, and acceleration offers insight into how objects move and react to stimuli in their environment.

Now, let’s delve into how we calculate velocity when a force acts on our puck. This involves using the formula \( v_f = v_i + at \) where \( v_i \) is the initial velocity, \( a \) is the acceleration from the applied force, and \( t \) represents time.

1. **Rightward Force:**
The initial velocity \( v_i \) of the puck is 3.00 m/s. With a rightward force, \( a = 156.25 \text{ m/s}^2 \) over \( t = 0.050 \text{ s} \), the final velocity \( v_f \) becomes: \[v_f = 3.00 + (156.25 \times 0.050) = 10.8125 \text{ m/s}\] Thus, the puck ultimately moves at 10.81 m/s to the right.2. **Leftward Force:**
When the force reverses direction, \( a = -75.0 \text{ m/s}^2 \) yields: \[v_f = 3.00 + (-75.0 \times 0.050) = -0.75 \text{ m/s}\] Here, the puck's velocity indicates a movement to the left at 0.75 m/s post impact.

These calculations unveil how forces alter the puck's speed and direction, giving a fuller picture of the applied principles of kinematics and dynamics.