a constant power p is applied on a particle of mass m.find the kinetic energy ,velocity and displacement of particle as function of time t

When a constant power \( P \) is applied to a particle of mass \( m \), we can derive the expressions for kinetic energy, velocity, and displacement as functions of time \( t \). Let's break this down step by step.

Understanding Power and Its Relation to Work

Power is defined as the rate at which work is done. Mathematically, it can be expressed as:

P = \frac{dW}{dt}

Where \( W \) is the work done. When a constant power is applied, we can integrate this relationship over time to find the work done on the particle.

Finding Work Done Over Time

Since power is constant, we can express the work done \( W \) over a time interval \( t \) as:

W = P \cdot t

Relating Work to Kinetic Energy

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. If we assume the particle starts from rest, the initial kinetic energy \( KE_0 = 0 \). Thus, the kinetic energy \( KE \) at time \( t \) can be expressed as:

KE = W = P \cdot t

Velocity as a Function of Time

The kinetic energy is also related to velocity by the equation:

KE = \frac{1}{2} m v^2

Setting the two expressions for kinetic energy equal gives us:

P \cdot t = \frac{1}{2} m v^2

From this, we can solve for velocity \( v \):

v = \sqrt{\frac{2Pt}{m}}

Displacement as a Function of Time

To find displacement \( s \), we can use the relationship between velocity and displacement. Since velocity is the derivative of displacement with respect to time, we can express this as:

v = \frac{ds}{dt}

Substituting our expression for velocity into this equation gives:

\frac{ds}{dt} = \sqrt{\frac{2Pt}{m}}

To find displacement, we integrate both sides with respect to time:

s = \int \sqrt{\frac{2Pt}{m}} \, dt

Carrying out this integration leads to:

s = \frac{2P}{3m} t^{3/2} + C

Assuming the initial displacement is zero (i.e., \( C = 0 \)), we have:

s = \frac{2P}{3m} t^{3/2}

Summary of Results

• Kinetic Energy: \( KE = Pt \)
• Velocity: \( v = \sqrt{\frac{2Pt}{m}} \)
• Displacement: \( s = \frac{2P}{3m} t^{3/2} \)

These equations provide a comprehensive understanding of how a particle behaves under constant power. Each variable is interrelated, showcasing the fundamental principles of physics in motion.