in the horizontal plane is captured by a camera attached to a balloon. The horizontal positions were recorded at \(0.1 \mathrm{~s}\) intervals: \begin{tabular}{l|l|l|l|l|l|l} \hline\(t(\mathrm{~s})\) & \(0.0\) & \(0.1\) & \(0.2\) & \(0.3\) & \(0.4\) & \(0.5\) \\ \hline\(x(\mathrm{~m})\) & \(10.00\) & \(11.00\) & \(11.75\) & \(12.25\) & \(13.00\) & \(14.00\) \\ \hline\(y(\mathrm{~m})\) & \(15.0\) & \(15.00\) & \(15.50\) & \(16.50\) & \(17.00\) & \(17.00\) \\ \hline \end{tabular} (a) Draw the motion diagram and the displacements for this motion. (b) Use the motion diagram to find the average velocity vectors. (c) Use the motion diagram to find the average acceleration vectors. (d) When is the speed and the acceleration maximum?

Step by step solution

01

Understand the Data

We have horizontal positions recorded at 0.1-second intervals. We need to use this data to draw a motion diagram and calculate the average velocity and acceleration vectors.

02

Plot the Motion Diagram

Plot the positions \(t, x, y\) on a graph with time (t) on the x-axis. Connect the points \(x,y\) over time to visualize the motion.

03

Calculate Displacements

Calculate the displacements (changes in position) at each interval: \[\Delta x = x_{i+1} - x_i \] and \[\Delta y = y_{i+1} - y_i \].

04

Draw Displacement Vectors

Draw vectors from each point to the next using the displacements calculated.

05

Calculate Average Velocity Vectors

Find the average velocity in each interval: \[\vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \frac{(\Delta x, \Delta y)}{0.1 \text{ s}} \].

06

Calculate Average Acceleration Vectors

Calculate the average acceleration at each interval using the change in velocity: \[\vec{a} = \frac{\Delta \vec{v}}{\Delta t} \].

07

Determine Maximum Speed and Acceleration

Find the magnitude of each velocity vector to determine speed \(|\vec{v}| = \sqrt{v_x^2 + v_y^2}\). Compare to find the maximum speed. Similarly, find the magnitude of acceleration \(a_x, a_y\) and compare.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Velocity

Average velocity is a measure of how quickly an object changes its position in a specific direction over a given time interval. It is a vector quantity, meaning it has both magnitude and direction. To calculate the average velocity vector for each interval, you can use the displacement and the time interval. The formula is:
\(\text{Average Velocity} = \frac{\text{Displacement}}{\text{Time Interval}}\).
Here's a breakdown:

• First, calculate the displacements in both the x and y directions using: \(\triangle x = x_{i+1} - x_i\) and \(\triangle y = y_{i+1} - y_i\).
• Next, plug these displacements into the average velocity formula: \(\frac{(\triangle x, \triangle y)}{0.1 \text{ s}}\).

This gives you the average velocity vector for each interval. In everyday terms, this tells you the average speed and direction of the moving object over each 0.1 second time interval.

Average Acceleration

Average acceleration is how quickly an object changes its velocity. Like velocity, it's also a vector, having both magnitude and direction. The average acceleration vector for each interval can be found using the change in velocity over the time interval. The formula is:
\(\text{Average Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Interval}}\).
Here's how to proceed:

• First, calculate the velocities at each time interval using the average velocity formula discussed before.
• Next, find the change in velocity (\triangle \ \ \ ) \(v\) between successive intervals.
• Plug this change into the acceleration formula \(\frac{\triangle v}{0.1 \text{ s}}\).

This step-by-step procedure allows you to determine how rapidly the velocity is changing at each interval, providing the average acceleration vector.

Displacement

Displacement is the straight-line change in position of an object from its initial to its final position. It is also a vector quantity, represented as \(\triangle \ x\) and \(\triangle \ y\) for each time interval. To find displacement, you can use:

• Calculate the difference in the x-coordinates between consecutive data points: \(\triangle \ x = x_{i+1} - x_i\).
• Do the same for the y-coordinates to get \(\triangle \ y\) \( = y_{i+1} - y_i\).

Connect these displacement vectors from one position to the next. This gives you a clearer picture of the object's overall path. Unlike distance, which is scalar and only considers the total ground covered, displacement focuses on the direction and final change in position.

Motion Diagram

A motion diagram helps visualize an object's movement over time by representing its position at various time intervals. Here's how to create one:

• Draw a graph with time (t) on the x-axis and positions (x, y) on the y-axis.
• Plot each recorded position as a point on this graph.
• Connect these points using lines or arrows to indicate the object's motion.
• Label each point with the corresponding time.
• Draw displacement vectors between each position to reinforce the direction and magnitude of movement.

This visual tool makes it easier to interpret the object's motion, see patterns in the movement, and understand how displacement, velocity, and acceleration interconnect. It's particularly useful when analyzing complex movements, making it simpler to comprehend how the object changes position over time.