Position, Velocity, and Acceleration vs Time Graphs

When an object moves with constant velocity (meaning zero acceleration), the relationship between position, velocity, and acceleration graphs follows a simple but important pattern. Let's explore what these three graphs look like when there is no acceleration.

The Setup

Consider an object moving at a constant velocity $v = 3\text{ m/s}$, starting from position $x_0 = 0\text{ m}$. Since the velocity is constant, the acceleration $a = 0\text{ m/s}^{2}$ throughout the motion.

We can describe the motion mathematically:

• Position: $x(t) = x_0 + vt = 3t$
• Velocity: $v(t) = 3\text{ m/s}$ (constant)
• Acceleration: $a(t) = 0\text{ m/s}^{2}$ (zero)

Key Characteristics

Notice these important features when acceleration is zero:

1. Position graph: Shows a straight line with a constant slope. This slope equals the velocity.

2. Velocity graph: Appears as a horizontal line, showing the velocity remains unchanged.

3. Acceleration graph: Also a horizontal line, but at zero.

Important Relationships

The graphs are connected in meaningful ways:

• The slope of the position graph equals the value shown on the velocity graph
• The slope of the velocity graph (which is zero) equals the value on the acceleration graph

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