a car starts from rest and accelerates such that instantaneous power remains constant the instantaneous velocity of this car is proportional to time as

To understand how a car that starts from rest accelerates with constant instantaneous power, let's break down the concepts involved.

Power and Velocity Relationship

Instantaneous power (\(P\)) is defined as the rate at which work is done or energy is transferred. Mathematically, it can be expressed as:

• P = F * v

Where:

• P is the power (in watts)
• F is the force (in newtons)
• v is the instantaneous velocity (in meters per second)

Since the car starts from rest, its initial velocity is zero. As the car accelerates, we want to keep the power constant. This means that as the velocity increases, the force must adjust accordingly to maintain that constant power output.

Acceleration and Force

Using Newton's second law, we know that force can also be defined as:

• F = m * a

Where:

• m is the mass of the car (in kilograms)
• a is the acceleration (in meters per second squared)

By substituting this into the power equation, we can express power in terms of mass and acceleration:

• P = m * a * v

Proportional Relationship between Velocity and Time

If we assume that the instantaneous power remains constant while the car accelerates, we can derive a relationship between velocity and time. With constant power, we can rearrange our equation:

• a = P / (m * v)

Since acceleration \(a\) is the derivative of velocity with respect to time (\(a = dv/dt\)), we can rewrite the equation:

• dv/dt = P / (m * v)

This is a separable differential equation. By rearranging and integrating, we find that:

• ∫v * dv = ∫(P/m) * dt

Upon integrating, we obtain:

• (1/2)v² = (P/m)t + C

Where \(C\) is the integration constant, which can be determined based on the initial conditions (the car starts from rest, so at \(t=0\), \(v=0\), thus \(C=0\)).

Now we can express the velocity as a function of time:

• v(t) = sqrt((2P/m)t)

Conclusion on Proportionality

From this equation, we can see that the instantaneous velocity \(v\) is indeed proportional to the square root of time \(t\). This means that as time progresses, the velocity of the car increases in a way that relates to the constant power output. Therefore, we conclude that while the instantaneous power remains constant, the relationship leads to a scenario where the instantaneous velocity increases as the square root of time, demonstrating a unique aspect of motion under constant power conditions.