Kinetic Energy and Work

wagonA.gif (13591 bytes)


Previously Asked Questions

Q:     "A 1000 kg car travels on a frictionless surface at a speed of 3.00 m/sec.  It is momentarily brought to rest as it compresses a spring in its path.  The spring constant k is 1200 N/m.  What distance d is the spring compressed?"

A:     To solve this problem we need to use the work-kinetic energy theorem:

 delta - delta.gif (839 bytes)K = Kfinal - Kinitial = W.

The work done by a spring force is given by the equation:

Ws = - 1 over 2 - onehalf.gif (67 bytes) k x2.

In this case, the work done by the spring force on the car as the spring is compressed a distance d from its rest state is given by:

Ws = - 1 over 2 - onehalf.gif (67 bytes) k d2

This is the work done by the spring force. 
Now, we need to find the change in kinetic energy of the car.  The car's initial kinetic energy is given by the equation:

  Kinitial = 1 over 2 - onehalf.gif (67 bytes) mv2

When the car stops moving its kinetic energy, which is the energy of motion, is zero:

Kfinal = 0.

Therefore, the change in the car's kinetic energy (delta - delta.gif (839 bytes)K) is:

delta - delta.gif (839 bytes)K = Kfinal - Kinitial  =  0 - 1 over 2 - onehalf.gif (67 bytes) mv2  =  - 1 over 2 - onehalf.gif (67 bytes) mv2

delta - delta.gif (839 bytes)K- 1 over 2 - onehalf.gif (67 bytes) mv2

We can now set the two quantities equal to each other (delta - delta.gif (839 bytes)K = Ws)  as required by the work-kinetic energy theorem.

delta - delta.gif (839 bytes)K = Ws  -> - 1 over 2 - onehalf.gif (67 bytes)mv2 = - 1 over 2 - onehalf.gif (67 bytes) k d2  -> mv2 = k d2

mv2 = k d2

Solve for the distance d the spring is compressed:

mv2 = k d2  -> d2 = mv2/k -> d = v square root - sqrt.gif (76 bytes)(m/k)

d = v square root - sqrt.gif (76 bytes)(m/k)  = (3.00 m/sec) square root - sqrt.gif (76 bytes)(1000 kg/1200 N/m)  = 2.74 m.

So the spring will be compressed 2.74 meters by the time the car comes to a stop.

Note: These equations could also be used to find the spring constant if the distance is given.  In which case, k = mv2 / d2.

Q:     What is Kinetic Energy?

A:     Kinetic Energy is the energy of motion.  Any object that moves has some energy due to the fact that its moving.  This energy is equal to half of the object mass multiplied by its velocity squared.  Since mass and velocity squared are never negative, kinetic energy is also never negative.

Q:     What is work?

A:     Work is energy transferred from or to an object via a force acting on that object.  Energy transferred to the object is positive work, likewise energy transferred away from the object is negative work.  Work is the dot product of the force and the displacement of the object.

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References

Units

   SI unit:  joule.gif (366 bytes)
    Atomic scale unit:  1 electron-volt = 1 eV = 1.60 X 10-19 J

 

Relevant Equations

Kinetic Energy K = onehalf.gif (67 bytes) mv2
Work - Kinetic Energy Theorem delta.gif (839 bytes)K = Kfinal - Kinitial = W
                 or
Kfinal  =  Kinitial  + W
Work done by a constant force W = Fd cos phi2.gif (845 bytes) = F . d
Work done by constant net force 7-13.gif (279 bytes)
Change in KE due to the total work delta.gif (839 bytes)K = Kfinal - Kinitial = W1 + W2 + W3 + . . .
Work done by weight Wg = mgd cos phi2.gif (845 bytes)
Work done in lifting and lowering and object delta.gif (839 bytes)K = Kfinal - Kinitial = Wa + Wg
Work done by variable force 7-31.gif (1151 bytes)
Hooke's Law (for spring forces) F = - kd
Hooke's Law is spring lies along the x-axis F = -kx
Work done by a spring force 7-40.gif (317 bytes)
Work done by a spring force if xinitial = 0 and xfinal = x Ws = - onehalf.gif (67 bytes) k x2
Power 7-44.gif (155 bytes)
Instantaneous Power 7-45.gif (168 bytes)
Instantaneous Power if F is at angle phi2.gif (845 bytes) to the direction of travel of the object P = Fv cos phi2.gif (845 bytes)F . v
Relativistic Kinetic Energy 7-51.gif (473 bytes)

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Mechanics List of Topics

Measurements Newton's Laws Potential Energy and Conservation of Energy Rotation of
Rigid Bodies
Elasticity
Vectors Forces and Fields Linear Momentum Angular Momentum Mechanical
Oscillations
Motion of Point-Mass Objects in One Dimension The Gravitational Field Collisions Torque Mechanical Waves
Motion of Point-Mass Objects in Two and Three Dimensions Kinetic Energy
and Work
Circular Motion of Point-Mass Objects Equilibrium Sound

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