A particle of mass is thrown upward with velocity v and there is regarding air resistance proportional to the square of the velocity with proportionality constant k. If the particle attains a maximum height after time t, and g isbthe gravitational acceleration, what is the velocity v?

Question

Answer

Understanding the Problem
To analyze the motion of a particle thrown upward with air resistance, we need to consider the forces acting on it:
- Gravitational Force: Acts downward with a magnitude of mg.
- Air Resistance: Acts downward as well, proportional to the square of the velocity, represented as kv^2.
Net Force and Motion Equation
The net force (F_net) acting on the particle can be expressed as:
F_net = -mg - kv^2
Using Newton's second law, we can relate this to acceleration (a):
ma = -mg - kv^2
This gives us the equation of motion:
a = -g - (k/m)v^2
Velocity as a Function of Time
The velocity of the particle changes as it moves upward. At the maximum height, the velocity becomes zero. Therefore, we can set up the following equation:
dv/dt = -g - (k/m)v^2
To find the initial velocity (v) that allows the particle to reach maximum height after time t, we can integrate the above equation.
Finding the Initial Velocity
Using the known conditions:
- At t = 0, v = v (initial velocity).
- At t = t, v = 0 (maximum height reached).
The integration will yield a relationship between the initial velocity v, time t, and constants g and k. The derived formula will help us find the necessary initial velocity.
Conclusion
The initial velocity v necessary for the particle to reach its maximum height after time t, considering air resistance proportional to the square of the velocity, can be computed using the equations of motion and the integration process outlined. This approach combines kinematics and dynamics to derive the desired velocity effectively.

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