An object of mass, m is moving with a constant velocity, v. How much work should be done on the object in order to bring the object to rest?
Approach Solution - 1
When an object is moving with a constant velocity, its kinetic energy is given by the equation:
Kinetic Energy= \(\frac 12mv^2\)
If you want to bring the object to rest, you need to remove all of its kinetic energy. The work done (W) is equal to the change in kinetic energy. In this case, the change in kinetic energy is equal to the initial kinetic energy (when the object is moving with velocity v).
So, the work done (W) to bring the object to rest is given by:
W= Change in Kinetic Energy = Final Kinetic Energy − Initial Kinetic Energy
Since the final kinetic energy is zero (the object is at rest), and the initial kinetic energy is \(\frac 12mv^2\), the work done is:
\(W=0−\frac 12mv^2\)
Therefore, the work done to bring the object to rest is \(-\frac 12mv^2\). The negative sign indicates that work is done against the motion of the object, removing its kinetic energy.
Approach Solution -2
When an object is brought to rest from a constant velocity, work must be done to overcome the object's kinetic energy and reduce it to zero. The work done W to bring an object of mass m moving with a constant velocity v to rest is equal to the change in kinetic energy.
The kinetic energy KE is given by the formula:
\(KE = \frac{1}{2} m v^2\)
When the object comes to rest, its final velocity \(v_f\) is 0, so the change in kinetic energy \(\Delta KE\) is:
\(\Delta KE = KE_{\text{final}} - KE_{\text{initial}}\)
\(\Delta KE = 0 - \frac{1}{2} m v^2\)
\(\Delta KE = -\frac{1}{2} m v^2\)
Since work is the transfer of energy, the work done on the object is equal to the change in kinetic energy:
\(W = \Delta KE\)
\(W = -\frac{1}{2} m v^2\)
So, the work done on the object to bring it to rest is \(-\frac{1}{2} m v^2\) or \(\frac{-1}{2}\) times its initial kinetic energy. The negative sign indicates that the work done is in the opposite direction of the object's initial motion.
So, the answer is: \(-\frac{1}{2} m v^2\)
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Concepts Used:
Work
Work is the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
Work Formula:
W = Force × Distance
Where,
Work (W) is equal to the force (f) time the distance.
Work Equations:
W = F d Cos θ
Where,
W = Amount of work, F = Vector of force, D = Magnitude of displacement, and θ = Angle between the vector of force and vector of displacement.
Unit of Work:
The SI unit for the work is the joule (J), and it is defined as the work done by a force of 1 Newton in moving an object for a distance of one unit meter in the direction of the force.
Work formula is used to measure the amount of work done, force, or displacement in any maths or real-life problem. It is written as in Newton meter or Nm.