FIGURE EX2.12 shows the velocity-versus-time graph for a particle moving along the x-axis. Its initial position is at x₀ = 2 m at t₀ = 0 s. What are the particle's position, velocity, and acceleration at t = 1.0 s?

Verified step by step guidance

1. Step 1: Analyze the velocity-versus-time graph. From the graph, the velocity at t = 1.0s is constant and equal to 1 cm/s (0.01 m/s). This indicates that the particle is moving at a constant velocity during this time interval.
2. Step 2: Use the formula for position when velocity is constant: x = x₀ + v * t. Here, x₀ = 2 m (initial position), v = 0.01 m/s (velocity at t = 1.0s), and t = 1.0s. Substitute these values into the formula to calculate the position.
3. Step 3: Since the velocity is constant during the interval from t = 0s to t = 3s, the acceleration is zero. Acceleration is defined as the rate of change of velocity, a = (Δv)/(Δt). For t = 1.0s, Δv = 0, so a = 0 m/s².
4. Step 4: Summarize the results: At t = 1.0s, the particle's position can be calculated using the formula in Step 2, its velocity is 0.01 m/s, and its acceleration is 0 m/s².
5. Step 5: To verify, observe the graph again. The flat line from t = 0s to t = 3s confirms constant velocity, and no change in velocity means zero acceleration during this interval.

Key Concepts

Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude and direction, and is typically expressed in units such as meters per second (m/s). In the context of the velocity-versus-time graph, the value of velocity at any point indicates how fast and in which direction the particle is moving.

Acceleration

Acceleration is the rate of change of velocity over time, indicating how quickly an object is speeding up or slowing down. It is also a vector quantity and is measured in units like meters per second squared (m/s²). On a velocity-time graph, the slope of the line represents the acceleration; a steeper slope indicates greater acceleration.

Position from Velocity-Time Graph

The position of a particle can be determined from its velocity-time graph by calculating the area under the velocity curve. Each segment of the graph corresponds to a specific time interval, and the area represents the displacement during that interval. By integrating the velocity over time, one can find the total change in position from the initial position.