Question

which velocity vs. time graph corresponds to the data represented by the position vs. time graph?
position (m)
time (s)
velocity vs time

Explanation:

To determine the corresponding velocity - time graph from a position - time graph, we use the relationship between position, velocity, and time. The velocity \(v\) is the slope of the position - time (\(x - t\)) graph, that is \(v=\frac{\Delta x}{\Delta t}\).

Step 1: Analyze the position - time graph

The given position - time graph is a straight line passing through the origin. For a straight - line \(x - t\) graph, the slope \(\frac{\Delta x}{\Delta t}\) is constant. Mathematically, if we consider two points \((t_1,x_1)\) and \((t_2,x_2)\) on the \(x - t\) graph, the slope \(m=\frac{x_2 - x_1}{t_2 - t_1}\) is constant. Since \(x\) is directly proportional to \(t\) ( \(x = vt\) where \(v\) is constant), the velocity \(v\) is constant.

Step 2: Analyze the velocity - time graph implications

A constant velocity means that in the velocity - time (\(v - t\)) graph, the velocity should be represented as a horizontal line (because the velocity does not change with time). The given velocity - time graph in the problem has a non - constant velocity (it has increasing, constant, and decreasing phases), but if we were to find the correct one, it should be a horizontal line (constant velocity) corresponding to the constant slope of the position - time graph.

However, since the problem is about matching the position - time graph (with constant slope, hence constant velocity) to a velocity - time graph, the correct velocity - time graph should show a constant velocity (a horizontal line). If we assume that the options (even though not fully shown) have a graph with a horizontal line (constant velocity), that would be the correct one.

If we consider the general case, for a linear \(x - t\) graph (constant slope), the \(v - t\) graph is a horizontal line (constant velocity).

Answer:

The velocity - time graph corresponding to the given position - time graph (which has a constant slope) should be a horizontal line (representing constant velocity). If we assume the options include a graph with a horizontal line (constant velocity), that is the correct one. (Since the provided velocity - time graph in the problem is not the correct one for the given \(x - t\) graph, and the correct \(v - t\) graph for a linear \(x - t\) graph is a horizontal line showing constant velocity.)