A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in m/s) of the particle is zero, the speed (in m/s) after 5 s is ?

To determine the speed of the particle after 5 seconds when it is acted upon by a constant power, we can use the relationship between power, force, and velocity. Let's break this down step by step.

Understanding Power and Work

Power is defined as the rate at which work is done or energy is transferred. In this case, the power delivered to the particle is constant at 0.5 watts. The relationship between power (P), force (F), and velocity (v) is given by the formula:

P = F × v

Finding Force

First, we need to consider how force relates to the acceleration of the particle. The net force acting on the particle can be determined using Newton's second law:

F = m × a

where m is the mass of the particle and a is its acceleration. Since we can express the power in terms of force and velocity, we can set up an equation:

P = m × a × v

Acceleration and Velocity Relationship

From the definition of acceleration, we can also express it as:

a = dv/dt

Substituting this into the power equation gives us:

P = m × (dv/dt) × v

Solving for Speed

Given that the power is constant, we can integrate the expression over time to find the speed of the particle. Since we know the power (0.5 W) and mass (0.2 kg), we can rearrange the equation to find the velocity as a function of time:

Rearranging gives us:

dv = (P / m) * dt / v

Integrating both sides gives us:

∫dv = ∫(P/m) * dt

This leads us to:

v(t) = sqrt((2 * P / m) * t)

Calculating the Speed After 5 Seconds

Now we can substitute the values into this equation to find the velocity after 5 seconds:

P = 0.5 W

m = 0.2 kg

t = 5 s

Plugging in these values:

v(5) = sqrt((2 * 0.5 / 0.2) * 5)

This simplifies to:

v(5) = sqrt((1 / 0.2) * 5) = sqrt(5 * 5) = sqrt(25) = 5 m/s

Final Result

Therefore, the speed of the particle after 5 seconds is 5 m/s.