Use the Work-Energy Theorem to show that an object with initial velocity vo will travel a distance d across a rough horizontal surface before stopping, where d = vo2/(2mKg).
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To use the Work-Energy Theorem to derive the expression for the distance an object travels across a rough horizontal surface before coming to a stop, we need to consider the work done by the force of friction.
Let's start by defining some variables:
- vo is the initial velocity of the object
- v is the final velocity of the object (which is zero, since it comes to a stop)
- d is the distance traveled by the object before stopping
- m is the mass of the object
- K is the coefficient of kinetic friction between the object and the rough surface
According to the Work-Energy Theorem, the work done on an object is equal to the change in its kinetic energy. In this case, since the object comes to a stop, the change in kinetic energy is equal to the initial kinetic energy.
The initial kinetic energy of the object is given by:
KE_initial = 1/2 * m * vo^2
Now, let's evaluate the work done by the friction force. The work done by a constant force is given by the formula:
Work = Force * Distance * cos(theta)
In this case, the force is the force of friction, the distance is d (the distance traveled before stopping), and the angle theta is the angle between the force and the direction of displacement (which is 0 degrees for a horizontal surface).
The friction force can be calculated using the equation:
Friction force = μ * Normal force
where μ is the coefficient of kinetic friction and the Normal force is the force exerted perpendicular to the surface by the object.
Since the object is on a horizontal surface, the Normal force is equal to its weight:
Normal force = m * g
where g is the acceleration due to gravity.
Therefore, the work done by the friction force is:
Work = (μ * Normal force) * d * cos(0) = μ * m * g * d
According to the Work-Energy Theorem, the work done by the friction force is equal to the change in kinetic energy, so:
μ * m * g * d = KE_initial
Substituting the expression for KE_initial, we have:
μ * m * g * d = 1/2 * m * vo^2
Simplifying and rearranging the equation, we get:
d = vo^2 / (2 * μ * g)
Since the coefficient of kinetic friction is denoted by K instead of μ in the question, we can substitute K = μ * g into the equation:
d = vo^2 / (2 * K * g)
Finally, since g = 9.8 m/s^2 is the acceleration due to gravity, we can substitute g into the equation to get the final expression:
d = vo^2 / (2 * K * 9.8)
Therefore, the distance traveled by an object before stopping on a rough horizontal surface is given by:
d = vo^2 / (2 * K * 9.8)
Please note that the derivation above assumes a constant coefficient of kinetic friction throughout the object's motion and neglects other factors like air resistance or other forces acting on the object.