Passage VI

Two solid spheres, X and Y, have the properties listed in Table 1 below.

The spheres are simultaneously released just below the surface of the water in a large container (see Figure 1).

Three students make predictions about which sphere, if either, will be first to reach the bottom of the container.

Student 1

The net downward, force on a sphere, F_net, equals the weight of the sphere minus the strength of the buoyancy force exerted on the sphere by the water it displaces. The sphere that has the greater F_net will reach the bottom first. The strength of the buoyancy force on a sphere equals the weight of the water displaced by the sphere. However, because both X and Y have small volumes, the strength of the buoyancy force on each sphere will be much less than the sphere's weight, so the buoyancy force will have little effect on how fast the sphere sinks. Thus, due to Y's greater weight, Y will sink faster than X, and Y will reach the bottom of the container first.

Student 2

After the spheres are released, gravity will cause each sphere to accelerate until it reaches its terminal speed, v_t, the speed at which its weight equals the frictional force retarding its motion. Because v, is proportional to the square root of the ratio of a sphere's mass to its radius, and because the ratio for X is equal to the ratio for Y, the spheres will reach the same v_t. However, because Y has more mass than X, Y will have the greater acceleration, so Y will reach v_t first. Therefore, because Y will fall faster than X until X reaches its terminal speed, Y will have the greater average speed and will reach the bottom of the container first.

Student 3

Because the gravitational acceleration of a falling object is independent of the object's mass, X and Y will have the same acceleration. Moreover, because the ratio of mass to radius for X is equal to the ratio of mass to radius for Y, the spheres will reach the same v_t. Therefore,X and Y will have the same average speed and will reach the bottom at the same time.