Calculating Average Velocity with Constant Acceleration

The problem asks us to find the average velocity of an object that starts from rest and moves with constant acceleration over a specific time interval. The object begins its motion at \(x=0\) at \(t=0\), has an initial velocity of \(u=0\) (since it starts from rest), and a constant acceleration of \(a = 3 \, \text{m/s}^2\).

Understanding Average Velocity

Average velocity is defined as the total displacement of an object divided by the total time taken for that displacement. Mathematically, it is expressed as:

\(\text{Average Velocity} = \frac{\text{Displacement}}{\text{Time Interval}}\)

In this problem, we need to find the average velocity between \(t = 4 \, \text{s}\) and \(t = 8 \, \text{s}\).

Finding Position at Specific Times

Since the object starts from rest (\(u=0\)) at \(x=0\) at \(t=0\) and moves with constant acceleration \(a\), its position \(x(t)\) at any time \(t\) can be found using the kinematic equation:

\(x(t) = ut + \frac{1}{2}at^2\)

Given \(u=0\), the equation simplifies to:

\(x(t) = \frac{1}{2}at^2\)

We need to find the position at \(t_1 = 4 \, \text{s}\) and \(t_2 = 8 \, \text{s}\).

• Position at \(t_1 = 4 \, \text{s}\):
  \(x_1 = x(4 \, \text{s}) = \frac{1}{2}(3 \, \text{m/s}^2)(4 \, \text{s})^2 = \frac{1}{2} \times 3 \times 16 = 3 \times 8 = 24 \, \text{m}\)
• Position at \(t_2 = 8 \, \text{s}\):
  \(x_2 = x(8 \, \text{s}) = \frac{1}{2}(3 \, \text{m/s}^2)(8 \, \text{s})^2 = \frac{1}{2} \times 3 \times 64 = 3 \times 32 = 96 \, \text{m}\)

Calculating Displacement

The displacement (\(\Delta x\)) between \(t_1 = 4 \, \text{s}\) and \(t_2 = 8 \, \text{s}\) is the difference between the final position and the initial position in that interval:

\(\text{Displacement } (\Delta x) = x_2 - x_1\)
\(\Delta x = 96 \, \text{m} - 24 \, \text{m} = 72 \, \text{m}\)

Calculating Time Interval

The time interval (\(\Delta t\)) is the difference between the final time and the initial time:

\(\text{Time Interval } (\Delta t) = t_2 - t_1\)
\(\Delta t = 8 \, \text{s} - 4 \, \text{s} = 4 \, \text{s}\)

Calculating Average Velocity

Now we can calculate the average velocity using the definition:

\(\text{Average Velocity} = \frac{\text{Displacement}}{\text{Time Interval}} = \frac{\Delta x}{\Delta t}\)
\(\text{Average Velocity} = \frac{72 \, \text{m}}{4 \, \text{s}} = 18 \, \text{m/s}\)

Alternative Method: Average of Instantaneous Velocities

For motion with constant acceleration, the average velocity over a time interval is also equal to the average of the initial and final instantaneous velocities during that interval. The instantaneous velocity \(v(t)\) at any time \(t\) is given by \(v(t) = u + at\). Since \(u=0\), \(v(t) = at\).

• Instantaneous velocity at \(t_1 = 4 \, \text{s}\):
  \(v_1 = v(4 \, \text{s}) = (3 \, \text{m/s}^2)(4 \, \text{s}) = 12 \, \text{m/s}\)
• Instantaneous velocity at \(t_2 = 8 \, \text{s}\):
  \(v_2 = v(8 \, \text{s}) = (3 \, \text{m/s}^2)(8 \, \text{s}) = 24 \, \text{m/s}\)

The average velocity over the interval is:

\(\text{Average Velocity} = \frac{v_1 + v_2}{2} = \frac{12 \, \text{m/s} + 24 \, \text{m/s}}{2} = \frac{36 \, \text{m/s}}{2} = 18 \, \text{m/s}\)

Both methods yield the same result, confirming our calculation.

Conclusion

The average velocity of the object between time 4 s and 8 s is 18 m/s.

Revision Table: Key Concepts

Additional Information: Motion with Constant Acceleration

Motion with constant acceleration is a fundamental concept in kinematics. It describes situations where the velocity of an object changes at a steady rate. The key equations governing such motion are:

1. \(v = u + at\) (Velocity-Time relation)
2. \(s = ut + \frac{1}{2}at^2\) (Position-Time relation)
3. \(v^2 = u^2 + 2as\) (Velocity-Displacement relation)

Where:

• \(v\) is the final velocity
• \(u\) is the initial velocity
• \(a\) is the constant acceleration
• \(t\) is the time taken
• \(s\) is the displacement

In this problem, the object starts from rest (\(u=0\)) and the initial position is set as the origin (\(x=0\)), simplifying these equations.

Average velocity can be different from instantaneous velocity. Instantaneous velocity is the velocity at a specific point in time, while average velocity is the overall velocity over an interval. In cases of constant acceleration, the average velocity over an interval is exactly the average of the instantaneous velocities at the beginning and end of that interval, as demonstrated in the solution.