  # Kinetic Energy and Work

### 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: K = Kfinal - Kinitial = W. The work done by a spring force is given by the equation: Ws = - 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 = - 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 = 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 ( K) is: K = Kfinal - Kinitial  =  0 - mv2  =  - mv2 K =  - mv2 We can now set the two quantities equal to each other ( K = Ws)  as required by the work-kinetic energy theorem. K = Ws  -> - mv2 = - 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 (m/k) d = v (m/k)  = (3.00 m/sec) (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.

## References

### Units

SI unit: Atomic scale unit:  1 electron-volt = 1 eV = 1.60 X 10-19 J

### Relevant Equations

 Kinetic Energy K = mv2 Work - Kinetic Energy Theorem K = Kfinal - Kinitial = W                  or Kfinal  =  Kinitial  + W Work done by a constant force W = Fd cos = F . d Work done by constant net force Change in KE due to the total work K = Kfinal - Kinitial = W1 + W2 + W3 + . . . Work done by weight Wg = mgd cos Work done in lifting and lowering and object K = Kfinal - Kinitial = Wa + Wg Work done by variable force 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 Work done by a spring force if xinitial = 0 and xfinal = x Ws = - k x2 Power Instantaneous Power Instantaneous Power if F is at angle to the direction of travel of the object P = Fv cos =  F . v Relativistic Kinetic Energy # 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  ครั้งที่

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