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In physics, an average force refers to the constant force that would produce the same change in an object's motion as a variable force does over a specific time period. It is expressed through the equation of impulse \[ I = \overrightarrow{\mathbf{F}} \times t \] where \( I \) is the impulse, \( \overrightarrow{\mathbf{F}} \) is the average force, and \( t \) is the time over which the force is applied.

When determining average force, it is crucial to understand its relationship to changes in velocity and mass. For the problem at hand, knowing that one force is three times the other offers a direct path to calculating the necessary time intervals. Simplifying the relationship between forces allows solving for unknowns like the time interval \( t_2 \). This relationship is a key tool in analyzing and solving motion-related problems.

The ability to compute average force accurately helps solve real-world challenges, such as determining the force exerted by athletes during sports activities or calculating the impact forces in vehicle collisions.

The time interval \( t \) represents the duration over which an average force acts on an object. In the impulse-momentum theorem, this factor plays a significant role as it directly influences the total impulse produced by a force. The equation \[ I = \overrightarrow{\mathbf{F}} \times t \] clearly demonstrates how time interval affects the magnitude of impulse.

In problem-solving, understanding the time interval assists in analyzing how longer or shorter application of force impacts an object's momentum. For example, in our exercise, knowing that force \( \overrightarrow{\mathbf{F}}_{1} \) acts over a time of \( 3.2 \, \text{ms} \) enables us to determine the equivalent time interval \( t_2 \) for force \( \overrightarrow{\mathbf{F}}_{2} \) using the impulse equivalence.

Ultimately, the intuitive grasp of how time interval extends or reduces the effects of force is crucial. It also plays a role in engineering, where designing systems that withstand forces over time depends on precise timing calculations.

Problem solving in physics often involves breaking down complicated scenarios into manageable steps. The methodology includes the use of fundamental theorems, like the impulse-momentum theorem, to link known quantities and solve for unknowns. Here’s how the process typically unfolds:

• Identify what information is given and what is required.
• Write down relevant formulas and remember their relationships.
• Substitute known values into these formulas to uncover unknowns.
• Perform algebraic manipulations to isolate the desired variable.

The given problem elegantly exemplifies this process. By understanding the impact of average forces and time intervals, we solve for \( t_2 \). With force \( \overrightarrow{\mathbf{F}}_{1} \) being three times \( \overrightarrow{\mathbf{F}}_{2} \) and yielding the same impulse, solving the equation for different time intervals becomes straightforward using mathematical expressions.

Practicing this methodical way of thinking equips students to handle not only textbook problems but also real-world situations. Such skills are foundational in developing analytical thinking and practical application of physics concepts.