Figure 8-27 shows three plums that are launched from the same level with the same speed. One moves straight upward, one is launched at a small angle to the vertical, and one is launched along a frictionless incline. Rank the plums according to their speed when they reach the level of the dashed line, greatest first.

A figure which shows three situations involving a plane that is not frictionless and the block sliding along the plane

According to the principle of conservation of energy, the total energy of the plum is constant. At any point, the total energy of the plum is the sum of its kinetic energy and potential energy. Using this concept, the plums according to their speed can be ranked when they reach the level of the dashed line, greatest first.

Formula:

$K_i+U_i=W_f+U_f$

According to the principle of conservation of energy, the first plum which is projected vertically reaches a heightand with speed

$K_i+U_i=K_f+U_f$

As

role="math" localid="1657180083050" $\begin{array}{l}{K}_i=\frac{1}{m}{m}_i^2,\\ U_i=0,\\ K_f=\frac{1}{m}m{v}_f^2,\\ U_f=-mgh\\ \frac{1}{2}m{v}_i^2+0=\frac{1}{2}m{v}_f^2-mgh\\ \frac{1}{2}{v}_i^2+gh=\frac{1}{2}{v}_f^2\\ v_{f1}=\sqrt{v_0^2+2gh}\end{array}$

Similarly, for the second and third plum,

$v_{f2=}{v}_{f3}=\sqrt{v_0^2+2gh}$

Here we found that the final speed of all the three plums is independent of its mass as well as path.

Therefore, the rank of the plums according to their speed when they reach the level of the dashed line, the greatest first is $v_{f1}=v_{f2}=v_{f3}$.