Question

a student graphed the position of a cart during a 7 - second time interval. position vs. time graph which statement best describes the cart?

Explanation:

To solve this, we analyze the position - time graph:

Step 1: Recall the meaning of a position - time graph

In a position - time graph, the slope of the line represents the velocity of the object. The formula for slope (which is also the formula for velocity \(v\)) is \(v=\frac{\Delta x}{\Delta t}\), where \(\Delta x\) is the change in position and \(\Delta t\) is the change in time.

Step 2: Analyze the graph's slope

The graph is a straight line. A straight - line graph in a position - time plot indicates that the slope (and thus the velocity) is constant. Let's take two points on the graph. At \(t = 0\) s, the position \(x_0=2\) m. At \(t = 7\) s, the position \(x_7 = 5\) m (from the graph). Then \(\Delta x=x_7 - x_0=5 - 2 = 3\) m and \(\Delta t=7 - 0 = 7\) s. The velocity \(v=\frac{3}{7}\) m/s (a constant value). A constant velocity means the object is moving with uniform motion (constant speed in a straight line, since the graph is linear and position - time graph for straight - line motion with constant velocity is linear). If the slope were zero, the object would be at rest. If the slope were changing, the object would be accelerating. Since the slope is constant, the cart is moving with a constant velocity (or moving at a steady speed, or in uniform motion).

(Note: Since the question is about describing the cart's motion from the position - time graph, and we assume the options (not provided here but typical for such questions) would have options like "It is at rest", "It is moving with constant velocity", "It is accelerating" etc. The correct description is that the cart is moving with a constant velocity (or moving at a steady speed) because the position - time graph is a straight line, indicating uniform motion.)

If we assume the options are:
A. It is at rest.
B. It is moving with constant velocity.
C. It is accelerating.

The answer would be B. It is moving with constant velocity.

Answer:

To solve this, we analyze the position - time graph:

Step 1: Recall the meaning of a position - time graph

In a position - time graph, the slope of the line represents the velocity of the object. The formula for slope (which is also the formula for velocity \(v\)) is \(v=\frac{\Delta x}{\Delta t}\), where \(\Delta x\) is the change in position and \(\Delta t\) is the change in time.

Step 2: Analyze the graph's slope

The graph is a straight line. A straight - line graph in a position - time plot indicates that the slope (and thus the velocity) is constant. Let's take two points on the graph. At \(t = 0\) s, the position \(x_0=2\) m. At \(t = 7\) s, the position \(x_7 = 5\) m (from the graph). Then \(\Delta x=x_7 - x_0=5 - 2 = 3\) m and \(\Delta t=7 - 0 = 7\) s. The velocity \(v=\frac{3}{7}\) m/s (a constant value). A constant velocity means the object is moving with uniform motion (constant speed in a straight line, since the graph is linear and position - time graph for straight - line motion with constant velocity is linear). If the slope were zero, the object would be at rest. If the slope were changing, the object would be accelerating. Since the slope is constant, the cart is moving with a constant velocity (or moving at a steady speed, or in uniform motion).

(Note: Since the question is about describing the cart's motion from the position - time graph, and we assume the options (not provided here but typical for such questions) would have options like "It is at rest", "It is moving with constant velocity", "It is accelerating" etc. The correct description is that the cart is moving with a constant velocity (or moving at a steady speed) because the position - time graph is a straight line, indicating uniform motion.)

If we assume the options are:
A. It is at rest.
B. It is moving with constant velocity.
C. It is accelerating.

The answer would be B. It is moving with constant velocity.