91Ó°ÊÓ

Particle motion refers to the movement of a particle along a path in space. In the exercise, the particle is moving in a two-dimensional space, specifically in the \(x\)-\(y\) plane. The motion is influenced by the initial velocity given in vector form and a constant acceleration that affects only the \(y\) component.

Understanding particle motion involves:

• Analyzing the initial position and velocity to predict future movement.
• Applying the given acceleration to determine changes over time.

In our case, the particle's initial velocity is \( \mathbf{v} = 7.25 \mathbf{i} + 3.48 \mathbf{j} \ m/s \). This initial velocity sets the stage for how the particle will begin its journey, moving simultaneously in the horizontal and vertical directions.

Velocity is a vector quantity that describes the speed of a particle in a particular direction. It's crucial to distinguish velocity from speed, as velocity includes direction while speed does not.

Here, the initial velocity vector is broken into components:

• Horizontal Component: \( v_x = 7.25 \ m/s \)
• Vertical Component: \( v_y = 3.48 \ m/s \)

In kinematics, knowing these components allows us to calculate how fast and in which direction the particle is moving at any given time. Velocity can change over time when an acceleration is applied, as we will see with the effect of the given constant acceleration on the vertical component.

Constant acceleration means that the rate of change of velocity over time remains the same. In this problem, the particle experiences an acceleration only in the \(y\)-direction with \( \mathbf{a} = 0.85 \mathbf{j} \ m/s^2 \). This means that every second, the particle's velocity in the \(y\)-direction increases by 0.85 \( m/s \).

Using the formula:\[ v_y = v_{0y} + at \]we can calculate how the particle's velocity in the \(y\)-direction changes over time due to constant acceleration. Here:

• \( v_{0y} = 3.48 \ m/s \)
• \( a = 0.85 \ m/s^2 \)
• \( t \) is the time variable we're solving for.

This approach helps in predicting how the trajectory will adjust as time passes.

The change in trajectory direction is one of the fascinating aspects of motion. In this scenario, we're tasked with finding out how long it takes for the trajectory direction to change by a specific angle, \(30^{\circ}\).

Initially, the particle travels with a direction determined by its velocity components. The angle \( \theta \) is given by:\[ \theta_0 = \tan^{-1}\left(\frac{3.48}{7.25}\right) \]After acceleration impacts the \(y\)-velocity, the angle will change. The requirement is to find the time \( t \) when:\[ \theta_f = \theta_0 + 30^{\circ} \]By using the tangent function:\[ \tan(\theta_f) = \frac{v_y}{v_x} = \frac{3.48 + 0.85t}{7.25} \]we can solve for \( t \). In our completed calculations, \( t \approx 4.12 \) seconds.

• This change indicates a new direction in the particle's motion path.
• It shows the direct influence of sustained acceleration on trajectory.

Understanding trajectory direction change is essential for predicting particle paths in diverse fields like physics, engineering, and even animation.