Consider the following example:
A bullet of mass 45g is fired at a speed of 220 m/s into a 5.0 kg sandbag hanging from a string from the ceiling. The sandbag absorbs the bullet and begins to swing. To what maximum vertical height will it rise? (Assuming negligent air resistance and heat loss due to friction)
The way to solve this would be to calculate the kinetic energy of the bullet and then convert it to gravitational potential energy of the bullet + sandbag.
$$E_{k} = \frac12 * 0.045\,{\rm kg} * (220\,{\rm m/s})^2=1089\,{\rm J}$$
We could then find the gravitational potential energy here by substituting in $E_{p} = E_{k}$.
But, what I'm curious about here is, what would be the speed immediately after the bullet impacts the sandbag? Either the kinetic energy, or the momentum could be conserved, but not both.
$$P = 0.045\,{\rm kg} * 220 \,{\rm m/s} = 9.9 \,{\rm kg} \cdot \,{\rm m/s}$$
Then we find the speed after the collision of the sandbag + bullet to be:
$$9.9 \,{\rm kg} \cdot \,{\rm m/s} = 5.0\,{\rm kg} + 0.045\,{\rm kg} * X \,{\rm m/s}\to 1.96234 \,{\rm m/s} = X$$
So if momentum is conserved then the kinetic energy would be:
$$E_{k} = \frac12(5.0\,{\rm kg} + 0.045\,{\rm kg}) * (1.96324 \,{\rm m/s})^2=9.72\,{\rm J}$$
Should this question specifically state whether momentum or energy was conserved? How can you tell which one will be conserved?
Best Answer
Total momentum is always conserved, in both elastic and inelastic collisions, but total kinetic energy is only conserved in elastic collisions. This example seems to be a completely inelastic collision, because at the end the objects merge. There is a formula to calculate the final velocity $v$ of two object with speed $u_1$ and $u_2$ and mass $m_1$ and $m_2$ in a completely inelastic collision, which is: $$v=\frac{m_1u_1+m_2u_2}{m_1+m_2}$$ Here's a simple derivation:
since momentum is always conserved, the sum of momenta at the beginning is the same as the end: $$p_{i1}+p_{i2}=p_{f1}+p_{f2}$$
However, since this is a completely inelastic collision, at the end the two objects will merge, and so there will be only one final momentum. The final momentum is simply the sum of initial momenta, like final mass is the sum of initial masses: $$p_{1}+p_{2}=p_f\qquad m_1+m_2=m_f$$
Then: $$v=\frac{p_f}{m_f}\qquad v=\frac{p_1+p_2}{m_1+m_2}\qquad v=\frac{m_1u_1+m_2u_2}{m_1+m_2}$$
Total kinetic energy however is not conserved, as you can see summing initial kinetic energies and comparing with the final kinetic energy.