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When dealing with varying forces over time, integrating the force function can help find the change in momentum. In simpler terms, force integration involves calculating the area under the force-time curve, which gives us an impulse. Impulse is essentially the total push given to the object.

For a force that changes with time like the one given, you need to determine the integral of the force function over the time interval of interest. For the i-hat direction here, the force increases linearly with time, depicted by the function \(0.280 \; \text{N/s} \, t\). To find the impulse, or the change in momentum, you integrate this function between the specified times (0 to 2 seconds).

The integral of \(t\) with respect to \(t\) is \(0.5 t^2\). Calculating this from 0 to 2 gives:

• \(0.5 \times (2)^2 \times 0.280 \; \text{N/s} = 0.560 \; \text{N s}\)

This result gives the change in momentum in the i-hat direction due to the external force applied between 0 and 2 seconds.

Change in momentum, often referred to as \(\Delta p\), arises when an object is subjected to an external force. The initial momentum is altered by the impulse, the calculated result from force integration.

In our example problem, to find the final momentum, we add the change in momentum due to the external force to the initial momentum. For the i-hat component of momentum, the initial value is \(-3.00 \; \text{kg m/s}\), and the calculated change is \(0.560 \; \text{kg m/s}\). By adding these together:

• The final i-hat momentum becomes \( -3.00 + 0.560 = -2.44 \; \text{kg m/s}\).

Similarly, for the j-hat component, the change calculated was \(-1.20 \; \text{kg m/s}\), and the initial was \(4.00 \; \text{kg m/s}\). So,

• Final j-hat momentum is \(4.00 - 1.20 = 2.80 \; \text{kg m/s}\).

By knowing both components, we can find the total momentum at any given time.

External forces are influences from outside a system that change its momentum. They play a crucial role in determining how an object's motion evolves over time.

In this problem, a force acting on a box in both horizontal directions (i-hat and j-hat) was applied, modifying its momentum. The net effect of these forces over time is calculated via integration, as we did before.

The concept highlights that without such a force, an object would maintain its state of motion. However, with the presence of an external force like the ones here:

• A positive force in the i-hat direction increased momentum.
• A negative force in the j-hat direction decreased momentum.

External forces drive changes by providing energy that alters the object's velocity and, consequently, its momentum. This makes understanding these forces key to predicting and analyzing motion in physical systems.