A particle is projected upwards under gravity in a resisting medium whose resistance varies as tha score of the velocity find the motion of the particle.

To analyze the motion of a particle projected upwards under gravity while experiencing resistance that varies with the square of its velocity, we need to set up the equations of motion that govern this scenario. This involves understanding the forces acting on the particle and applying Newton's second law of motion.

Understanding the Forces at Play

When a particle is projected upwards, two main forces act on it:

• Gravitational Force: This force acts downward and is given by $F_g=mg$, where $m$ is the mass of the particle and $g$ is the acceleration due to gravity.
• Resistive Force: This force opposes the motion of the particle and varies with the square of the velocity, expressed as $F_r=-k{v}^2$, where $k$ is a constant that depends on the properties of the medium and $v$ is the velocity of the particle.

Setting Up the Equation of Motion

According to Newton's second law, the net force acting on the particle is equal to the mass of the particle multiplied by its acceleration:

\( F_{net} = ma = m \frac{dv}{dt}

Thus, we can write the equation of motion as:

\( m \frac{dv}{dt} = -mg - kv^2

Rearranging this gives us:

\( \frac{dv}{dt} = -g - \frac{k}{m}v^2

Separating Variables

To solve this differential equation, we can separate the variables:

\( \frac{dv}{-g - \frac{k}{m}v^2} = dt

This allows us to integrate both sides. The left side involves a more complex integral, which can be simplified using a substitution method.

Integrating the Left Side

Let’s perform the integration. The integral can be approached using partial fractions or a trigonometric substitution, depending on the form. For simplicity, we can express it as:

\( \int \frac{dv}{-g - \frac{k}{m}v^2}

This integral can be solved to yield a function of time, which will help us find the velocity as a function of time.

Finding Velocity and Position

After integrating, we can express the velocity $v(t)$ in terms of time. Once we have $v(t)$, we can find the position $s(t)$ by integrating the velocity function:

\( s(t) = \int v(t) dt

Behavior of the Particle

The solution will reveal how the particle's velocity decreases over time due to the combined effects of gravity and air resistance. Initially, the particle will ascend until its velocity reaches zero, after which it will begin to descend. The presence of the quadratic resistance means that the particle will not return to its original launch height, as energy is lost to the resistive force.

Conclusion

In summary, the motion of a particle projected upwards in a medium with resistance proportional to the square of its velocity can be modeled using differential equations derived from Newton's laws. By solving these equations, we can gain insights into the velocity and position of the particle over time, illustrating the effects of gravity and air resistance on its motion.