Q4.13 Energy conservation .... in disguise (S)

The y component of the velocity of a projectile moving under gravity satisfies

\[ v^2 = v_0^2 + 2 a(y-y_0) \]

with a=-g.

Show that this equation can be viewed as an expression of energy conservation.

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Hint

Try multiplying through by m.
 

Solution

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Solution

  • If we take the equation, multiply by m amd substitute a=-g
    \[ mv^{2}=mv_{0}^{2}+2ma(y-y_{0})=mv_{0}^{2}-2mgy+2mgy_{0} \]
  • Rearrange a bit:
    \[ mv^{2}+2mgy=mv_{0}^{2}+2mgy_{0} \]
  • Divide by 2:
    \[ \frac{1}{2}mv^{2}+mgy=\frac{1}{2}mv_{0}^{2}+mgy_{0} \]
    which expresses the conservation of the sum of kinetic and gravitational potential energies.