A particle at t₁ = ― 2.0 s is at 𝓍₁ = 5.2 cm and at t₂ = 3.4 s is at 𝓍₂ = 8.5 cm. What is its average velocity over this time interval? Can you calculate its average speed from these data? Why or why not?

Verified step by step guidance

Step 1: Recall the formula for average velocity, which is defined as the change in position (displacement) divided by the change in time. Mathematically, it is expressed as:

v

_avg = ΔxΔt, where

Δ x = x

₂ - x₁ and

Δ t = t

₂ - t₁.

Step 2: Substitute the given values into the formula for displacement:

Δ x = 8.5 \, cm - 5.2 \, cm

. Similarly, calculate the time interval:

Δ t = 3.4 \, s - (-2.0 \, s)

Step 3: Use the calculated values of

Δ x

and

Δ t

to find the average velocity:

v

_avg = ΔxΔt. Ensure the units are consistent (e.g., convert cm to m if needed).

Step 4: To determine if the average speed can be calculated, recall that average speed is defined as the total distance traveled divided by the total time. From the given data, we only know the initial and final positions, not the path taken by the particle. Therefore, we cannot calculate the total distance traveled unless additional information about the particle's trajectory is provided.

Step 5: Conclude that while the average velocity can be calculated using the displacement and time interval, the average speed cannot be determined from the given data because the total distance traveled is unknown.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity, meaning it has both magnitude and direction. In this case, it can be calculated by finding the difference in position (𝓍₂ - 𝓍₁) and dividing it by the time interval (t₂ - t₁). This provides insight into the overall change in position of the particle over the specified time.

Average Speed

Average speed is the total distance traveled divided by the total time taken, and it is a scalar quantity, meaning it only has magnitude and no direction. To calculate average speed, one must consider the total path length covered by the particle, regardless of direction. In this scenario, since the particle moves from 5.2 cm to 8.5 cm, the average speed can be determined by the distance traveled over the time interval.

Displacement vs. Distance

Displacement refers to the change in position of an object and is a vector quantity, while distance is the total length of the path traveled and is a scalar quantity. In this problem, the displacement is the difference between the final and initial positions, while the distance would be the same in this case since the particle moves in one direction. Understanding this distinction is crucial for correctly calculating average velocity and average speed.