91Ó°ÊÓ

In physics, force and acceleration are intricately connected. When a force acts on an object, it causes the object to accelerate. This fundamental relationship is described by Newton's Second Law, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: \[ \vec{F} = m \vec{a} \]Here, \( \vec{F} \) represents the force vector, \( m \) is the mass, and \( \vec{a} \) is the acceleration vector.
For the given exercise, the force \( \vec{F} \) is provided as a vector function in terms of \( t \) (time):\[ \vec{F} = k_1 \hat{\imath} + k_2 t^3 \hat{\jmath} \]Breaking it down:

• The \( \hat{\imath} \) component, \( k_1 \), is constant.
• The \( \hat{\jmath} \) component varies with time as \( k_2 t^3 \).

Using Newton's Second Law, we can find acceleration by dividing each component of the force by \( m \):\[ \vec{a} = \frac{k_1}{m} \hat{\imath} + \frac{k_2 t^3}{m} \hat{\jmath} \]This gives us a clear expression of how the object's acceleration will vary over time based on the exerted forces.

Once we have the acceleration, the next step is to find velocity. Velocity is essentially the antiderivative of acceleration. In other words, we integrate the acceleration function \( \vec{a}(t) \) with respect to time \( t \) to determine velocity \( \vec{v}(t) \).
The integration process involves dealing with each component of acceleration separately:

• For the \( \hat{\imath} \) component:\[ \int \frac{k_1}{m} \, dt = \frac{k_1}{m} t + C_1 \]
• For the \( \hat{\jmath} \) component:\[ \int \frac{k_2 t^3}{m} \, dt = \frac{k_2}{4m} t^4 + C_2 \]

These results combine to give the general expression for velocity:\[ \vec{v}(t) = \left( \frac{k_1}{m} t + C_1 \right) \hat{\imath} + \left( \frac{k_2}{4m} t^4 + C_2 \right) \hat{\jmath} \]The constants \( C_1 \) and \( C_2 \) arise due to the indefinite integration and are determined using initial conditions.

Initial conditions provide the necessary information to ascertain the specific trajectory of an object's motion. In this problem, the object starts from rest, meaning its initial velocity at time \( t = 0 \) is zero, denoted by \( \vec{v}(0) = 0 \).By setting \( t = 0 \) in the general velocity equation, we solve for the constants:

• \( \hat{\imath} \) component:\[ 0 = \frac{k_1}{m} \times 0 + C_1 \]This implies \( C_1 = 0 \).
• \( \hat{\jmath} \) component:\[ 0 = \frac{k_2}{4m} \times 0 + C_2 \]This implies \( C_2 = 0 \).

Substituting these values back, the specific function for velocity as a function of time is:\[ \vec{v}(t) = \frac{k_1}{m} t \hat{\imath} + \frac{k_2}{4m} t^4 \hat{\jmath} \]Thus, by applying initial conditions, we tailor the general solution to match the unique state of the system at the beginning.