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Impulse is a key concept in physics, especially when analyzing how forces influence the motion of objects. In simple terms, impulse is the product of a force applied to an object and the time duration over which it is applied. Mathematically, impulse (\( J \)) is described as an integral over a force-time graph, which can be expressed as:\[ J = \int_{t_0}^{t_1} \vec{F}(t) \, dt \]This can be understood as the total change in momentum of an object when subjected to a varying or constant force. The direction of the impulse is the same as the direction of the applied force. In our exercise, a force function \( \vec{F}(t) = 100 e^{-2t} \hat{\mathrm{i}} \)is applied, showcasing a decreasing exponential force over time. The aim is to compute the impulse when this force acts from \( t = 0 \) to \( t = 2 \) seconds. By solving the integral:\[ J = \int_{0}^{2} 100 e^{-2t} \, dt \]we find that the impulse is approximately \( 49.01 \, \mathrm{Ns} \hat{\mathrm{i}} \). This result reflects the total effect of the force on the particle's momentum over the specified interval.

When studying motion under the influence of variable forces, it's often useful to calculate the average force over a given time interval. The average force \( \vec{F}_{\text{avg}} \)is defined as the total impulse divided by the total time interval \( \Delta t \). Mathematically, it is given as:\[ \vec{F}_{\text{avg}} = \frac{1}{\Delta t} \int_{t_0}^{t_1} \vec{F}(t) \, dt \]This expression enables us to understand the overall effect of a varying force as if it were constant throughout the time interval. In the given exercise, since the impulse calculated was \( 49.01 \, \mathrm{Ns} \hat{\mathrm{i}} \)and the time interval was \( 2 \) seconds, the average force becomes:\[ \vec{F}_{\text{avg}} = \frac{49.01}{2} \, \mathrm{N} \hat{\mathrm{i}} \approx 24.51 \, \mathrm{N} \hat{\mathrm{i}} \]This approximation provides insight into the force's effective strength over the duration, helping understand the motion of the particle as if it were subject to a constant \( 24.51 \) Newtons over \( 2 \) seconds.

Integral calculus is a branch of mathematics that is essential for understanding physics concepts like impulse and average force. It involves finding the integral of functions which represent accumulation of quantities, such as the total force applied over time. The definite integral \( \int_{a}^{b} \)is used to calculate the net effect of a varying function, such as a force, between two limits; in this case, from \( t=0 \) to \( t=2 \).Here, we deal with the force function \( \vec{F}(t) = 100 e^{-2t} \). By integrating this function over the limits 0 to 2,we effectively sum up all the infinitesimally small pieces of force applied over each tiniest segment of time. The integral\[ \int_{0}^{2} 100 e^{-2t} \, dt \]translates to calculating the area under the force-time curve. This solution utilizes properties of exponential functions and integral calculus methodologies to derive an exact value for impulse, which then contributes to finding the average force.Integral calculus, therefore, not only aids in solving problems involving changing velocities and forces but also serves as the foundation for more complex physics topics, illustrating the link between mathematical theory and physical phenomena.