The basic definition of work is . This form of the work allows us to obtain a quantity that has significant meaning in physics. That is, starting from Newton's second law of motion, , the work done by the force along the displacement is

 

In the last equality, the acceleration and displacement are considered in the same direction; therefore, their scalar product is the same as the algebraic product of their magnitudes (hyperlink to general calculus). The expression for the work can be re-written using the relation for uniform accelerated motion, . From where, . Thus, the work is

In the previous expression, the effect of work is to change the quantity to . This quantity is called the kinetic energy of the object moving at the velocity ,

Kinetic Energy, :

 

Notices that the kinetic energy of an object is always a positive quantity and independent of the direction of motion.

In addition, the kinetic energy of an object is not an absolute property of the object. In fact, the kinetic energy of an object is relative to the observer's frame of reference.

For example, in the drawing on the left, if observer A throws a ball inside the bus with a velocity , the corresponding kinetic energy of the ball, with respect to the people sitting in the bus, is . Since observer B sees the same ball moving at the velocity , where is the velocity of the bus with respect to observer B, he will assign to the ball a kinetic energy given by . These two energies are clearly different for each observer; being the kinetic energy of the ball measured by observer B greater than the energy  measured by observer A.

At the same time, the effect of ball hitting an object inside the bus is clearly weaker than the effect of the ball hitting an object outside the bus. Thus, this simple example illustrate that the kinetic energy is relative to the observer.

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Unit of Kinetic Energy

The unit of Kinetic Energy is the same as the unit of Work. In fact,

Unit of Kinetic Energy = (Kilogram) (Meter/Second)2 = (Kilogram) (Meter/Second2) (Meter) = (Newton) (Meter)

Unit of Kinetic Energy = Joule

This result should be expected because the difference between the final and initial kinetic energy is the work and only quantities of the same nature can be added or subtracted.

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Work- Energy Theorem

The mathematical expression shown above, ,  is called the work-energy theorem because this formula establishes the relation between the work and the kinetic energy,

This relation can be read in different forms,

  1. If work is done by the external force in the same direction as the displacement, the final kinetic energy of the object is greater than the original kinetic energy of the object.

  2. if work is done by the external force in opposite direction as the displacement, the final kinetic energy of the object is smaller than the original kinetic energy of the object.

  3. If the kinetic energy of an object increases, work is done on the object by an external force.

  4. if the kinetic energy of an object decreases, the object does work on the rest of the system. Therefore, the kinetic energy of an object is the capacity of the object to perform work because of its motion.

  5. If the net work is zero, the kinetic energy is unchanged. This is the case of an object being lifted from the floor to a table when the object starts at rest and with zero kinetic energy, and the object ending with zero kinetic energy.

The change in kinetic energy is an observable that can be measured experimentally for all classical objects; then, the change in kinetic energy is the result of all forces acting on the object such as external agent applying forces, frictional forces, normal force, or gravitational force. Thus,

The change in Kinetic Energy  of a system is equal to the Net Work acting on the system even when the actual forces acting on the system may be various or only part of one of the forces is the responsible of the change (see application below).

 

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Stopping Distance versus Initial Velocity

 

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The graph on the left shows the necessary stopping distance for a vehicle originally moving at different velocities when the force applied by the brakes is constant and not reaction time (or reaction distance) is considered in the analysis. In this case, the equations of motion are

where the angle between the displacement of the car and the external applied force (in this instance, the force of friction with the road) is 1800. Moreover, at the end of the braking, the car is at rest with the final velocity, , equals to zero. Therefore, the above equation is reduced to from where . For the graph on the left, the ratio between the mass and the force is one-tenth, . Notices that the independent coordinate (horizontal) shown in the graph is the velocity while the dependent variable (vertical) is the displacement.

Analyzing the relation, , it can be observed that the stopping distance may depend on the mass; that is, an increment on the mass of the car produce a linear increment on the stopping distance of the car. However, the frictional force also depend on the mass of the object, , where is the coefficient of friction between the surfaces. If it is an object sliding over another surface, it is the coefficient of kinetic friction. On the other side, if it is a wheel turning over a surface, it is the coefficient of static friction assuming that the maximum braking occurs when the static friction is maximum. In either case, the stopping distance becomes .

 
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Example

 

  1) Question

 

a)

-19.3 KJ

 

b)

-490.0 KJ

 

c)

55.7 KJ

 N

d)

-29.5 KJ

 

e)

None of the above.

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com