Question

In each part, the velocity versus time curve is given for a particle moving along a line. Use the curve to find the displacement and the distance traveled by the particle over the time interval \(0 \leq \mathrm{t} \leq 3\).

Short answer

Answer: Displacement is the difference between the initial and final positions of a particle, while distance traveled is the total length of the path the particle moves through over a given time interval. Displacement can be determined by calculating the area under the velocity versus time curve (integral of the velocity function over the time interval), while distance traveled can be determined by calculating the area under the curve while considering the absolute value of the velocity function (integral of the absolute value of the velocity function over the time interval). Displacement and distance traveled can be compared to understand the direction(s) of a particle's motion during the given time interval.

Step-by-step solution

Understand displacement and distance traveled

Displacement is the difference between the initial and final positions of a particle, while the distance traveled is the total length of the path the particle moves through over a given time interval. In this case, we will use the provided velocity versus time curve to determine these values.

Analyze the provided velocity versus time curve

Without the actual curve provided, we cannot offer specific analysis. However, when analyzing a given curve, pay attention to any zero crossings and their significance in terms of displacement and distance traveled. It is also important to note whether the curve is above or below the time axis, as this will affect the integrals when calculating displacement and distance.

Calculate displacement using the integral of the velocity curve

Displacement can be found by calculating the area under the velocity versus time curve, as displacement is the integral of velocity with respect to time. For the given time interval \(0 \leq \mathrm{t} \leq 3\), find the integral of the given velocity function over the interval (denoted here as V(t)). Displacement can be represented as: \(Displacement = \int_{0}^{3} V(t) \, dt\) Evaluate this integral for the provided velocity curve to determine the displacement.

Calculate distance traveled by considering the absolute value of velocity

Distance traveled can also be found by calculating the area under the velocity versus time curve, but taking the absolute value of the velocity function to consider the length of the path traveled, not just the net change in position. For the given time interval \(0 \leq \mathrm{t} \leq 3\), find the integral of the absolute value of the given velocity function over the interval (denoted here as V(t)). Distance traveled can be represented as: \(Distance\,Traveled = \int_{0}^{3} |V(t)| \, dt\) Evaluate this integral for the provided velocity curve to determine the distance traveled.

Interpret the results

Once you have calculated the displacement and distance traveled, compare these values to understand the particle's motion. If displacement and distance traveled are equal, the particle has moved in only one direction during the given time interval. However, if the distance traveled is greater than the displacement, the particle has changed directions during the given time interval.