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An energy analysis begins by first identifying which types of forces are doing net work upon the object. In the case of the projectile, frictional forces (air resistance) are said to be negligible. The only remaining force acting upon the projectile is gravity - a conservative force. The force of gravity is the only force doing work upon the projectile.
Since the work being done on the projectile is being done by a conservative force, the total mechanical energy will be conserved. See Know the Law section. The amount of total energy - KE plus PE - at the beginning of the motion will be the same amount at every location along the path of the projectile.
The goal will be to determine the amount of energy at one of the locations so that the energy at all locations can be determined. The obvious starting point is location A since the height and speed is known for this location. Once the KE and PE of the projectile are determined for location A (see Formula Frenzy section), the total amount of energy will be known. Heights at locations B and C can be used to determine the PE at these locations. The KE simply makes up the difference between the total energy and the PE. And of course at location D, there is no potential energy since the projectile is at ground level (or a pico-meter above); all the energy is in the form of KE.
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Energy Conservation:
If the only forces doing net work upon an object are conservative forces (such as gravity and spring forces), then the mechanical energy of the object will be conserved. The energy may change from one form to another - potential to kinetic or vice versa; but the total amount of the two forms together will be unchanging.
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The amount of kinetic energy (KE) possessed by an object depends upon its mass (m) and its velocity (v). The formula is
KE = 0.5 • m • v2
The amount of potential energy (PE) possessed by an object depends upon its mass (m) and its height (h). The formula is
PE = m • g • h
where g is the gravitational field strength (9.8 N/kg on Earth).
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