the speed v reached by a car of mass m in travelling a distance x starting from rest,driven with constant power P,is given by??

To determine the speed \( v \) reached by a car of mass \( m \) traveling a distance \( x \) from rest while being driven with constant power \( P \), we can use the relationship between power, force, and speed. Let's break this down step by step.

Understanding Power and Motion

Power is defined as the rate at which work is done or energy is transferred. In the context of a car, the power \( P \) can be expressed as:

• P = F * v

Here, \( F \) is the force exerted by the car, and \( v \) is its velocity. Since the car starts from rest, we can relate the force to the mass and acceleration using Newton's second law:

• F = m * a

Connecting Power, Force, and Velocity

Substituting the expression for force into the power equation gives us:

• P = m * a * v

Now, we can express acceleration \( a \) in terms of velocity \( v \) and distance \( x \). Since the car starts from rest, we can use the kinematic equation:

• x = (1/2) * a * t^2

From this, we can derive acceleration as:

• a = \frac{2x}{t^2}

Relating Distance, Time, and Velocity

We also know that velocity can be expressed as:

• v = a * t

Substituting our expression for acceleration into this equation gives:

• v = \frac{2x}{t}

Combining the Equations

Now, substituting \( a \) back into the power equation:

• P = m * \frac{2x}{t^2} * \frac{2x}{t}

This simplifies to:

• P = \frac{4mx^2}{t^3}

From this, we can solve for time \( t \):

• t = \left(\frac{4mx^2}{P}\right)^{1/3}

Finding the Final Speed

Now that we have \( t \), we can substitute it back into our equation for velocity:

• v = \frac{2x}{t}

Substituting the expression for \( t \) gives:

• v = \frac{2x}{\left(\frac{4mx^2}{P}\right)^{1/3}}

This simplifies to:

• v = \frac{2x \cdot \left(\frac{P}{4mx^2}\right)^{1/3}}{1}

Finally, we can express the speed \( v \) as:

• v = \left(\frac{P}{m}\right)^{1/3} \cdot (2x)^{2/3}

Final Expression

In summary, the speed \( v \) reached by the car after traveling a distance \( x \) under constant power \( P \) is given by:

• v = \left(\frac{P}{m}\right)^{1/3} \cdot (2x)^{2/3}

This equation shows how the speed of the car depends on the power supplied, the mass of the car, and the distance traveled. Each component plays a crucial role in determining how quickly the car can accelerate.