91Ó°ÊÓ

Newton's second law is a fundamental principle in particle dynamics, relating the motion of a particle to the forces acting upon it. It is commonly expressed as \( \mathbf{F} = m \mathbf{a} \), where \( \mathbf{F} \) is the net force applied to a particle, \( m \) is the mass of the particle, and \( \mathbf{a} \) is the acceleration resulting from that force.
This law tells us that a particle's acceleration depends directly on the net force acting on it and inversely on its mass. In simpler terms, a particle with more mass will require more force for the same acceleration compared to a lighter particle. Moreover, any change in the applied force results in a corresponding change in acceleration.
In our exercise, the force acting on the particle changes with time, expressed by the function \( \mathbf{F} = (4 + 2t) \mathbf{i} + (t^2 - 2) \mathbf{j} + 5 \mathbf{k} \). By applying Newton's second law, we can determine the particle's acceleration by dividing the force by the mass, which is essential for solving the problem.

Acceleration describes the rate at which an object's velocity changes with time. It is a vector quantity, meaning it has both magnitude and direction. In the context of Newton's second law, acceleration \( \mathbf{a} \) is found by dividing the force \( \mathbf{F} \) by mass \( m \), resulting in \( \mathbf{a} = \frac{\mathbf{F}}{m} \).
In our given problem, the mass of the particle is \( 0.25 \mathrm{kg} \), and the forces are functions of time. Let’s break down the components:

• For the \( \mathbf{i} \) direction: \( \frac{4 + 2t}{0.25} = 16 + 8t \).
• For the \( \mathbf{j} \) direction: \( \frac{t^2 - 2}{0.25} = 4t^2 - 8 \).
• For the \( \mathbf{k} \) direction: \( \frac{5}{0.25} = 20 \).

With these components, the acceleration function becomes \( \mathbf{a} = (16 + 8t) \mathbf{i} + (4t^2 - 8) \mathbf{j} + 20 \mathbf{k} \ \mathrm{m/s^2} \). This indicates how the rate of change in velocity is shaped by the force components.

Integration is a mathematical technique used to determine the cumulative effect of a quantity over an interval, often used in motion dynamics to find changes in velocity or position. When we're given acceleration, integrating it over a time interval provides the velocity change \( \Delta \mathbf{v} \).
In this scenario, to determine the particle's velocity at \( t_2 = 4 \mathrm{s} \), we integrate the acceleration function \( \mathbf{a} \) from \( t_1 = 2 \mathrm{s} \) to \( t_2 = 4 \mathrm{s} \):

• For \( \mathbf{i} \): \( \int_{2}^{4} (16 + 8t) \ dt = 80 \).
• For \( \mathbf{j} \): \( \int_{2}^{4} (4t^2 - 8) \ dt = \frac{64}{3} \).
• For \( \mathbf{k} \): \( \int_{2}^{4} 20 \ dt = 40 \).

This integration provides the components of the velocity change, which can then be added to the initial velocity to find the final velocity of the particle.

Velocity is key in describing an object's motion, indicating the speed and direction at which an object is moving. It is also a vector quantity, with both magnitude and direction.
In our problem, the initial velocity at \( t_1 = 2 \mathrm{s} \) is given by \( \mathbf{v}_{1} = 2 \mathbf{i} + \mathbf{j} - \mathbf{k} \mathrm{m/s} \). After integrating the acceleration over the interval \( [2, 4] \), we find the change in velocity \( \Delta \mathbf{v} \).
Adding \( \Delta \mathbf{v} \) to \( \mathbf{v}_{1} \), we calculate the final velocity \( \mathbf{v}_2 \) at \( t_2 = 4 \mathrm{s} \):

• For \( \mathbf{i} \): \( 2 + 80 = 82 \).
• For \( \mathbf{j} \): \( 1 + \frac{64}{3} = \frac{67}{3} \).
• For \( \mathbf{k} \): \( -1 + 40 = 39 \).

Thus, the resulting velocity is \( \mathbf{v}_2 = 82 \mathbf{i} + \frac{67}{3} \mathbf{j} + 39 \mathbf{k} \ \mathrm{m/s} \). This indicates how the velocity has evolved due to the forces applied.

Force is a fundamental interaction in physics that causes an object to undergo a change in motion. As per Newton's second law, force is directly related to the acceleration of a particle through the equation \( \mathbf{F} = m \mathbf{a} \).
In the exercise, the force acting on the particle varies with time and is given by \( \mathbf{F} = (4+2t) \mathbf{i} + (t^2-2) \mathbf{j} + 5 \mathbf{k} \ \mathrm{N} \). Forces influence how a particle's velocity changes over time, impacting both direction and magnitude.
Breaking down the force:

• \( (4+2t) \mathbf{i} \) implies a time-dependent force along the x-axis.
• \( (t^2-2) \mathbf{j} \) implies a quadratic function of time affecting motion along the y-axis.
• \( 5 \mathbf{k} \) indicates a constant force in the z-axis direction.

This detailed force analysis is crucial for understanding the resultant acceleration and eventual velocity change of the particle within the defined time interval.