**Velocity of an Automobile on a Highway**The velocity of a fast automobile on a straight highway is given by the following function, where \( t \) is measured in seconds and \( v \) has units of meters per second (m/s):\[ v(t) = \begin{cases} 3t & \text{if } 0 \leq t < 25 \\75 & \text{if } 25 \leq t < 50 \\225 - 3t & \text{if } t \geq 50 \end{cases} \]### Explanation of the Velocity FunctionThe function \( v(t) \) describes the velocity of the automobile in different intervals of time:1. **For \( 0 \leq t < 25 \):** The velocity \( v(t) \) increases linearly with time according to the equation \( v(t) = 3t \). This means that for every second, the velocity increases by 3 meters per second. 2. **For \( 25 \leq t < 50 \):** The velocity \( v(t) \) is constant at 75 meters per second. This indicates that the automobile maintains a steady speed of 75 m/s during this time interval. 3. **For \( t \geq 50 \):** The velocity \( v(t) \) decreases linearly with time according to the equation \( v(t) = 225 - 3t \). This indicates that the velocity decreases by 3 meters per second every second.This piecewise function effectively describes how the automobile accelerates, maintains a constant speed, and then decelerates over successive intervals of time. ### Questions for Automotive Motion Analysis#### c. Distance Traveled Calculation**Question:** What is the distance traveled by the automobile in the first 75 seconds? #### d. Position Determination**Question:** What is the position of the automobile when \( t = 90 \) seconds?**Notes:** The above questions are aimed at analyzing the motion of an automobile over a given time interval. To address these questions, consider using kinematic equations or any provided velocity-time graphs or data tables related to the automobile's motion. Calculations should account for initial velocities, accelerations, and any pertinent information described in the broader problem set.##### Example Approach:- For distance calculation, integrate the velocity over the given time period.- For position determination, calculate the position using a known initial position and integrating velocity as a function of time up to the specified time of 90 seconds.These calculations play a crucial role in understanding the motion dynamics of an automobile and are fundamental in physics and engineering contexts.

Name: **Velocity of an Automobile on a Highway**The velocity of a fast auto
Uploaded: 2024-03-20T06:46:42.000Z
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Description: **Velocity of an Automobile on a Highway**The velocity of a fast automobile on a straight highway is given by the following function, where \( t \) is measured in seconds and \( v \) has units of meters per second (m/s):\[ v(t) = \begin{cases} 3t &