A 1.00 g grasshopper and a 0.100 g cricket leap vertically into the air with the same initial speed in the absence of air drag.

Which of the following statements must be true? (More than one statement may be true.)

A. The grasshopper goes 10 times as high as the cricket.
B. Both will reach the same height.
C. At the highest point, both will have the same amount of gravitational potential energy relative to the ground.
D. At the highest point, the gravitational potential energy for the grasshopper will be 10 times as great as that for the cricket.
E. The grasshopper and the cricket have the same initial kinetic energy.

To answer this question, we need to consider the conservation of energy principle.

The total mechanical energy of the system (the grasshopper or cricket) is conserved when there is no air drag acting on it. This means that the initial mechanical energy (kinetic energy + gravitational potential energy) will be equal to the final mechanical energy at any point during the motion.

Let's analyze each statement:

A. The grasshopper goes 10 times as high as the cricket.
To determine if this statement is true, we need to compare the heights reached by the grasshopper and the cricket. Since both have the same initial speed and are free from air drag, their initial kinetic energy is the same. The difference in their masses is given as 1.00 g for the grasshopper and 0.100 g for the cricket. As gravitational potential energy depends on mass and height, it does not imply that the grasshopper will go 10 times as high as the cricket. Thus, statement A is not necessarily true.

B. Both will reach the same height.
This statement is true because, regardless of their mass, if both the grasshopper and the cricket have the same initial speed and no air drag acts on them, they will reach the same height. This is due to the conservation of mechanical energy. So, statement B is true.

C. At the highest point, both will have the same amount of gravitational potential energy relative to the ground.
This statement is also true. At the highest point of their trajectory, both the grasshopper and the cricket come to a stop momentarily before falling back down. At this point, all their initial kinetic energy is converted to gravitational potential energy. Since their heights are the same (as mentioned in statement B), their gravitational potential energies will also be the same, being relative to the same reference point (the ground). So, statement C is true.

D. At the highest point, the gravitational potential energy for the grasshopper will be 10 times as great as that for the cricket.
This statement contradicts statement C, which we have already established as true. Therefore, statement D is not true.

E. The grasshopper and the cricket have the same initial kinetic energy.
This statement is true. It is mentioned earlier that both the grasshopper and the cricket leap into the air with the same initial speed. Since kinetic energy depends on mass and speed, and the given information does not specify any difference in their speeds, it implies they have the same initial kinetic energy. So, statement E is true.

In conclusion, statements B, C, and E are all true.