Question

An object moves along a line so that its velocity at time \(t\) is \(v(t)=3 t^{2}-24 t+36\) feet per second. Find the displacement and total distance traveled by the object for \(-1 \leq t \leq 9 .\)

Short answer

Displacement is 130 feet; total distance is 230 feet.

Step-by-step solution

Understand the Concept of Displacement and Total Distance

Displacement is the net change in position of the object over an interval of time, calculated by integrating the velocity function over that interval. Total distance traveled is the sum of all changes in position, which requires integrating the absolute value of the velocity function to account for any changes in direction.

Set Up the Integral for Displacement

To find the displacement, integrate the velocity function:\[\text{Displacement} = \int_{-1}^{9} v(t) \, dt = \int_{-1}^{9} (3t^2 - 24t + 36) \, dt.\]

Calculate the Displacement

Evaluate the integral:\[\int (3t^2 - 24t + 36) \, dt = t^3 - 12t^2 + 36t + C.\]Evaluate this from \(-1\) to \(9\):\[(9^3 - 12 \times 9^2 + 36 \times 9) - ((-1)^3 - 12 \times (-1)^2 + 36 \times (-1)).\]Simplify to get:\[(729 - 972 + 324) - (-1 - 12 - 36) = 81 - (-49) = 130.\]Thus, the displacement is 130 feet.

Identify Critical Points for Velocity

To find total distance, first find when the velocity is zero to locate potential changes in direction by solving \(3t^2 - 24t + 36 = 0\). Solveusing the quadratic formula:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{24 \pm \sqrt{576 - 432}}{6} = \frac{24 \pm 12}{6}.\]This gives roots at \(t = 6\) and \(t = 2\). These points divide the interval into sections where the velocity may change sign.

Calculate Total Distance with Absolute Value of Velocity

The total distance is found by integrating the absolute value of the velocity over the given intervals:\[\text{Distance} = \int_{-1}^{2} |v(t)| \, dt + \int_{2}^{6} |v(t)| \, dt + \int_{6}^{9} |v(t)| \, dt.\]Evaluate each integral considering the sign of velocity in each interval:1. \(t \in [-1, 2]: v(t) \geq 0\), so \( ext{integrate } v(t)\).2. \(t \in [2, 6]: v(t) \leq 0\), so \( ext{integrate } -v(t)\).3. \(t \in [6, 9]: v(t) \geq 0\), so \( ext{integrate } v(t)\).Integrate and sum the result to get the total distance:\[\(\int_{-1}^{2} (3t^2 - 24t + 36) \, dt = 50.667\)\]\[\int_{2}^{6} -(3t^2 - 24t + 36) \, dt = 160\]\[\int_{6}^{9} (3t^2 - 24t + 36) \, dt = 19.333\]Sum of the distances: \(50.667 + 160 + 19.333 = 230\) feet.

Key concepts

Displacement Calculation

When we talk about displacement in calculus, we're referring to the net change in the position of an object over a specific timeframe. Displacement is all about how far out of place an object is; it's the object's overall change in position. To calculate displacement, we integrate the velocity function over the given interval of time.

Imagine the velocity function as a speedometer reading over time. By integrating this function, you're capturing the total accumulation of motion, just like calculating the total distance driven using the speedometer. For the given velocity function, to find the displacement between time t=-1t=−1t=−1 and t=9t=9t=9, you set up an integral that looks like this:

• \[ \text{Displacement} = \int_{-1}^{9} (3t^2 - 24t + 36) \, dt \]

By solving this integral, you determine the net movement of the object, not considering any backtracking or changes in direction.

In this case, the computed result of 130 feet indicates the net distance moved from the initial position to the final position after the entire interval.

Velocity Integration

Velocity integration forms the foundation of analyzing motion in calculus. It involves integrating a velocity function, which represents the rate of change of an object's position. By performing this integration, you calculate the total displacement during a specified period. Let's see how it's applied:

• First, you identify the velocity function, which in this case is given by \( v(t) = 3t^2 - 24t + 36 \).
• The integration process translates this continuous rate of change into a concrete value of total displacement.

The significance of velocity integration becomes apparent when evaluating complex motion paths, enabling the determination of cumulative changes in position over time.

Approaching from a step-by-step perspective makes it straightforward: set up your integral, as mentioned earlier, solve it through evaluation, and you've captured the complete picture of displacement for a given time range.

Total Distance Traveled

Total distance differs from displacement in that it considers the entire path traveled by the object, taking into account any changes in direction. For a complete measurement of all motion, you integrate the absolute value of the velocity function.

Finding the total distance involves several steps:

• Identify points where the velocity changes sign, indicating a change in direction. These are critical points where the velocity function equals zero.
• Evaluate the integral over each interval considering the sign of the velocity. When velocity is negative, negate the function to account for movement in the opposite direction.

This approach ensures you account for every move forward and backward. For our example, the velocity changes direction at t=2t=2t=2 and t=6t=6t=6. So we split our integral into sub-intervals: \(\int_{-1}^{2} |v(t)| \, dt + \int_{2}^{6} |v(t)| \, dt + \int_{6}^{9} |v(t)| \, dt \).

By combining these positive and negative areas, we discover the full path has been 230 feet—much larger than the net 130 feet displacement, highlighting the back and forth of the journey.

Absolute Value Integration

Absolute value integration is a crucial concept when calculating total distance traveled, as it ensures all movement, whether forward or backward, is considered positively. Without using absolute values, backward movements might simply cancel out forward movements when calculating total displacement.

Here's why absolute value matters:

• It converts any negative portions of the velocity to positive, thus capturing the complete distance regardless of the direction of travel.
• This is essential when calculating the total length of the path, especially if the object reverses direction during its motion.

In summary, absolute value integration provides a more comprehensive measure of motion, ensuring every step and turn along the path contributes to the total distance. For complex trajectories, this approach accurately reflects the real-world scenario, where every journey’s entirety counts, no matter how circuitous it might be.