Question

A particle moving along the positive \(x\) axis has the following positions at various times: $$ \begin{array}{cccccccc} \hline x(\mathrm{~m}) & 0.080 & 0.050 & 0.040 & 0.050 & 0.080 & 0.13 & 0.20 \\\ t(\mathrm{~s}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \end{array} $$ (a) Plot displacement (not position) versus time. \((b)\) Find the average velocity of the particle in the intervals 0 to \(1 \mathrm{~s}, 0\) to 2 \(\mathrm{s}, 0\) to \(3 \mathrm{~s}, 0\) to \(4 \mathrm{~s}\). ( \(c\) ) Find the slope of the curve drawn in part \((a)\) at the points \(t=0,1,2,3,4\), and 5 s. \((d)\) Plot the slope (units?) versus time. \((e)\) From the curve of part \((d)\) determine the acceleration of the particle at times \(t=2,3\), and \(4 \mathrm{~s}\).

Short answer

The displacement was found to be negative initially, indicating a movement opposite to the positive X direction. The average velocities were found for 0s to 1s, 0s to 2s, 0s to 3s, and 0s to 4s. The instantaneous velocities were determined from the slopes at points \(t=0,1,2,3,4,5\). Acceleration values, derived from the velocity vs time plot, indicate how the velocity of the particle changes over time for \(t=2,3,4 s\).

Step-by-step solution

Displacement versus Time

First, the displacement for each time interval needs to be calculated. Displacement is the change in position, that is, final position minus the initial position. For example, the displacement from 0s to 1s is \(0.050m - 0.080m = -0.030m\). Repeat this process for each time interval and plot the results.

Average Velocity

Average velocity is calculated as the displacement divided by the time interval. For example, the average velocity from 0s to 1s is \(-0.030m / 1s = -0.030m/s\). Repeat this process for each of the intervals, from 0s to 2s, 0s to 3s, and 0s to 4s.

Slope at Each Point

The slope of a displacement-time curve at a specific point gives the instantaneous velocity at that point. To find the slope, draw a tangent line at each of the points \(t=0,1,2,3,4,5\) and calculate the slope of each tangent line. The slope can be approximated by calculating the ratio of the change in displacement over the change in time between two points close to the point of interest.

Plot Slope versus Time

After calculating the slope at each point (the instantaneous velocity), plot these values against time. This gives a graphical representation of how the velocity of the particle changes over time.

Acceleration

Acceleration is the rate of change of velocity with respect to time, which can be calculated from the slope of the velocity vs time curve made in Step 4. For times \(t=2,3,4 s\), calculate the slope of the curve (the change in velocity over the corresponding time).

Key concepts

Displacement

Displacement is all about how far an object moves from its starting point. It’s different from distance in that it considers direction too. To calculate displacement, simply subtract the initial position from the final position.
For example, when a particle moves from position 0.080 m to 0.050 m between time 0 s and 1 s, the displacement would be \(0.050 \, \text{m} - 0.080 \, \text{m} = -0.030 \, \text{m}\). Here, the negative sign indicates movement towards the starting point on the x-axis.
Calculating displacements over intervals helps to understand how an object's position changes with time, which is a key part of analyzing its motion. Tracking these changes and plotting them gives a clear visual indication of the overall journey of the particle.

Average Velocity

Average velocity provides a measure of how fast an object is moving over a certain time interval and in which direction. It's calculated by dividing the total displacement by the total time taken.
For example, between 0 s and 1 s, the average velocity is \(-0.030 \, \text{m} / 1 \, \text{s} = -0.030 \, \text{m/s}\). The negative sign here means the particle is moving in the opposite direction to the positive x-axis.
This concept is valuable for understanding the overall motion of an object over time, as it simplifies fluctuating movements into a single, average value. Calculating it for different intervals like between 0-2 s, 0-3 s, and so on, helps to compare how motion changes over time.

Instantaneous Velocity

Instantaneous velocity is like a snapshot of how fast and in what direction an object is moving at a specific moment in time. Unlike average velocity, which is over an interval, instantaneous velocity is at a point.
To find this, you would determine the slope of the tangent to the displacement-time curve at the point of interest. The slope is achieved by drawing a line that just touches the curve at the specific point and calculating the ratio of the change in displacement to the change in time.
Imagine taking a picture of a moving car; this is similar to finding the instantaneous velocity, as it pinpoints the exact velocity at exactly that "click" moment.

Acceleration

Acceleration tells us how quickly velocity is changing and involves both the magnitude and the direction of this change. It’s the rate of change of velocity with respect to time.
To calculate acceleration, one would find the slope of the velocity-time graph. This tells us by how much the velocity changes over a time interval. For instance, if you’re calculating acceleration at time 3s, you’d look at how the velocity changes between times say 2 to 4 s.
Understanding acceleration helps predict future motion and explains how quickly things are speeding up or slowing down. It's a crucial concept in kinematics, especially in situations where forces start acting on the object, influencing its speed and direction.