Question

image result for velocity vs. time graphs which of the velocity vs. time graphs above represents an object that begins with a positive velocity, accelerates negatively, and finally ends up moving at approximately the same speed as it started, but in the opposite direction? 2 4 1 6 5 3 image result for velocity vs. time graphs which of the velocity vs. time graphs above represents an object that begins with a positive velocity, slows to a complete stop, and accelerates back to the original velocity?

Explanation:

First Question (Object with positive initial velocity, negative acceleration, ends with same speed opposite direction)

Step1: Analyze initial velocity

The object starts with positive velocity, so the graph should begin above the time - axis (v > 0).

Step2: Analyze acceleration

Negative acceleration means the slope of the velocity - time graph is negative (since \(a=\frac{\Delta v}{\Delta t}\), negative acceleration implies a decreasing velocity over time when initial velocity is positive).

Step3: Analyze final velocity

The object ends up with the same speed but opposite direction. Speed is the magnitude of velocity, so if initial velocity is \(v_0\), final velocity is \(-v_0\) (same magnitude, opposite sign). So the graph should start at \(v = v_0\) (positive), have a negative slope (negative acceleration), and end at \(v=-v_0\) (negative, same magnitude as start).

Second Question (Object with positive initial velocity, slows to stop, accelerates back to original velocity)

Step1: Analyze initial velocity

Starts with positive velocity, so graph starts above time - axis.

Step2: Analyze slowing to stop

Slowing to a complete stop means velocity decreases to 0. So the graph should have a negative slope (since velocity is decreasing) until \(v = 0\).

Step3: Analyze accelerating back to original velocity

Accelerating back to the original velocity means velocity increases from 0 back to the initial positive velocity. So after reaching \(v = 0\), the graph should have a positive slope (since velocity is increasing) and end at the same positive velocity as the start.

Since the actual graphs are not fully visible (only options numbered 1 - 6 are shown with one option selected as 5 for the first part), but based on the description:

For the first question (positive initial velocity, negative acceleration, same speed opposite direction):
A graph that starts positive, has a negative slope, and ends at a negative velocity with the same magnitude as the start. If we assume the options are typical velocity - time graphs:

• A graph with a straight line starting at positive v, sloping down to negative v with the same magnitude (e.g., starts at \(v = 5\), ends at \(v=-5\) with negative slope) would be the one.

For the second question (positive initial velocity, slows to stop, accelerates back to original velocity):
A graph that starts positive, slopes down to 0 (slowing to stop), then slopes up back to the initial positive velocity (accelerating back).

If we consider the first question's selected option is 5, and assuming the graph for the first question (option 5) has the characteristics: starts positive, negative slope, ends negative with same speed.

Answer:

(for the first question as per the selected option):
5 (assuming option 5's graph has positive initial velocity, negative acceleration, and final velocity with same speed opposite direction)

For the second question, without seeing the full graph details, but based on the description, a graph that has a negative slope to 0 then positive slope back to original velocity would be the answer (e.g., if option 4 or another option has this shape, but since the image is partially visible, we can only provide the analysis - based answer structure as above).