91Ó°ÊÓ

Velocity is a fundamental concept in kinematics that tells us how fast something is moving in a specific direction. It is different from speed, which does not consider direction. Velocity is a vector quantity, meaning it has both magnitude and direction. In the context of the car problem, the initial velocity of the car is given as \(+36.0 \, \text{m/s}\). The positive sign indicates the direction of motion.
Knowing the velocity at different points in time helps us understand the car's motion throughout its journey. The final velocity after 18 seconds is \(+28.0 \, \text{m/s}\), and the problem asks us to find the velocity after the first 12 seconds, which is eventually calculated as \(+30.0 \, \text{m/s}\). To calculate velocity changes, we need to understand how velocity is impacted by acceleration and time.

Acceleration is the rate at which an object's velocity changes over time. Like velocity, it is a vector quantity, meaning it has both magnitude and direction. The average acceleration can show an increase or decrease in speed, or a change in motion direction.
The car in our problem experiences two distinct phases of acceleration: \(\overline{a}_1\) for 12 seconds, and \(\overline{a}_2\) for 6 seconds. It's known that \(\overline{a}_1 / \overline{a}_2 = 1.50\), which informs us about the relative strength between the two accelerations. By using the equations of motion, we determine that \(\overline{a}_2\) equals \(-\frac{1}{3} \, \text{m/s}^2\). A negative acceleration, like in this problem, suggests the car is slowing down or decelerating. This is why it's crucial to understand and recognize the signs associated with acceleration.

Kinematics heavily relies on motion equations to solve real-world problems. The main kinematic equation used here is: \[ v = v_0 + a \times t \]Where \(v\) is the final velocity, \(v_0\) is the initial velocity, \(a\) is acceleration, and \(t\) is time. This equation allows us to calculate the final velocity after a period of constant acceleration.
In this problem, for the first segment, we rearrange the equation to find \(v_1\) (the velocity at the end of 12 seconds): \[ v_1 = v_0 + \overline{a}_1 \times 12 \] For the second segment, the equation: \[ v_f = v_1 + \overline{a}_2 \times 6 \] allows us to link the velocity at the end of each phase. Solving the problem involves substituting values into these motion equations and exploiting relationships like \(\overline{a}_1 = 1.5 \times \overline{a}_2\) to find unknowns.
Understanding how to manipulate these equations is key in kinematics, allowing us to predict and analyze motion accurately.