The velocity graph of a particle moving along the x-axis is shown. The particle has zero velocity at t=0.00 s and reaches a maximum velocity, vmax, after a total elapsed time, ttotal. If the initial position of the particle is x0=7.28 m, the maximum velocity of the particle is vmax=33.4 m/s, and the total elapsed time is ttotal=20.3 s, what is the particle's position at t=13.5 s?

To find the particle's position at t = 13.5 s, we need to analyze the given information and use the concepts of velocity and position.

Given data:
Initial position (x₀) = 7.28 m
Maximum velocity (vₘₐₓ) = 33.4 m/s
Total elapsed time (t_total) = 20.3 s

To find the particle's position at t = 13.5 s, we need to break down the problem into two parts:

Part 1: Finding the displacement from t = 0.0 s to t = t_total
Part 2: Adding the displacement from Part 1 to the initial position (x₀) to find the final position at t = t

Part 1: Finding the displacement from t = 0.0 s to t = t_total

The velocity graph provides information about the particle's velocity at different times. Since the particle starts at rest (zero velocity), the area under the velocity-time graph from t = 0.0 s to t_total gives the displacement of the particle during this time.

In this case, the velocity-time graph appears to be a straight line, indicating constant acceleration. The area under a velocity-time graph represents the displacement. Since it is a triangle, we can find the area using the formula for the area of a triangle:

displacement = (base * height) / 2

The base of the triangle is t_total, and the height can be calculated using vmax.

height = vmax

Using the given values, we can calculate the displacement:
displacement = (t_total * vmax) / 2

Part 2: Adding the displacement from Part 1 to the initial position (x₀) to find the final position at t = t

To find the final position at t = t, we add the displacement from Part 1 to the initial position (x₀).

final position = x₀ + displacement

Substituting the values we obtained:

final position = 7.28 m + displacement

Now, let's calculate the displacement and find the final position at t = 13.5 s:

displacement = (20.3 s * 33.4 m/s) / 2

final position = 7.28 m + displacement