91Ó°ÊÓ

Velocity is a fundamental concept in physics that describes the rate at which an object changes its position. In this problem, the velocity of the particle is given by the vector \( \vec{v}=\left(8.00 \hat{i}+3.00 t^{2} \hat{j}\right) \mathrm{m/s} \). This equation means the particle's movement can be described by two components: a constant velocity in the \( x \)-direction (8.00 m/s) and a time-dependent velocity in the \( y \)-direction (3.00 \cdot t^2) m/s.

• The \( x \)-component of velocity, \( v_x = 8.00 \) m/s, indicates a steady motion in the horizontal direction.
• The \( y \)-component of velocity, \( v_y = 3.00 t^2 \) m/s, shows that the particle's velocity in the vertical direction increases as time passes.

This results in a path that curves, continuously changing the direction of travel. The velocity direction and magnitude at any given time can be understood by these components. Velocity, therefore, provides essential insight into how fast and in what direction a particle is moving, which is critical for predicting the object's future position.

Force, as explained through Newton's second law, represents the interaction that causes an object to accelerate. The relation shown by the law is \( \vec{F} = m \vec{a} \), where \( \vec{F} \) is force, \( m \) is mass, and \( \vec{a} \) is acceleration. In this problem, we are applying this concept to a \( 3.00 \) kg particle. Using the acceleration components from the velocity derivation:- The horizontal acceleration component \( a_x = 0 \) (derivative of constant \( 8.00 \) m/s is zero)- The vertical acceleration component \( a_y = 6.00 t \) (derivative of \( 3.00t^2 \) m/s is \( 6.00t \) m/s²)This leads to force components:

• \( F_x = m \cdot a_x = 0 \) N
• \( F_y = m \cdot a_y = 18.0 t \) N

Given the net force is \( 35.0 \) N and \( F_x = 0 \), we determine \( F_y = 35.0 \) N, ensuring the force is along the \( y \)-direction at this specific moment. Understanding how force components result from acceleration helps students grasp the real-life applications of Newton's second law, ensuring they see force as not merely an abstract concept but an actionable one affecting everyday physics.

Acceleration measures how quickly an object changes its velocity, typically as a result of an acting force. In this exercise, acceleration derives from the velocity's time-varying component. Particularly, we calculate the acceleration components by differentiating the velocity components with respect to time. The outcome is:

• \( a_x = 0 \) m/s², since the velocity in the \( x \)-direction is constant
• \( a_y = 6.00t \) m/s², since the velocity in the \( y \)-direction is \( 3.00t^2 \)

Acceleration is vital for understanding how objects move and interact in a physical space. It links directly to force, as described by Newton's second law. When students see that the acceleration is what dictates the varying component of force, they can better appreciate how acceleration's magnitude and direction are continually altering except when velocities are constant. Finally, by setting the calculated force equal to the given force, we found the time at which the particle's acceleration aligns with that force magnitude, closing the loop of understanding the dynamic nature of motions compelled by forces.