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 = K_{final} - K_{initial} = W. The work done by a spring force is given by the equation: W_{s} = - k x^{2}. 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: W_{s} = - k d^{2} This is the work done by the
spring force. K_{initial} = mv^{2} When the car stops moving its kinetic energy, which is the energy of motion, is zero: K_{final} = 0. Therefore, the change in the car's kinetic energy (K) is: K = K_{final} - K_{initial} = 0 - mv^{2} = - mv^{2} K = - mv^{2} We can now set the two quantities equal to each other (K = W_{s}) as required by the work-kinetic energy theorem. K = W_{s} -> - mv^{2} = - k d^{2} -> mv^{2} = k d^{2} mv^{2} = k d^{2} Solve for the distance d the spring is compressed: mv^{2} = k d^{2} -> d^{2} = mv^{2}/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 = mv^{2} / d^{2}. |
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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SI unit:
Atomic scale unit: 1 electron-volt = 1 eV = 1.60 X 10^{-19}
J
Kinetic Energy | K = mv^{2} |
Work - Kinetic Energy Theorem | K = K_{final} - K_{initial}
= W or K_{final} = K_{initial} + 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 = K_{final} - K_{initial} = W_{1} + W_{2} + W_{3} + . . . |
Work done by weight | W_{g} = mgd cos |
Work done in lifting and lowering and object | K = K_{final} - K_{initial} = W_{a} + W_{g} |
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 x_{initial} = 0 and x_{final} = x | W_{s} = - k x^{2} |
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 |
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