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Difficulty: HardGravity and Free Fall

Two solid spheres of identical volume and exterior finish—one cast from dense lead and the other from lightweight aluminum—are released simultaneously from a hovering atmospheric research drone. Which of the following best describes their descent characteristics through the air?

  1. The lead sphere will attain a higher terminal velocity because more air resistance is required to counteract its greater weight.Answer
  2. B
    Both spheres will accelerate at an identical rate throughout the entire fall because gravitational acceleration is independent of mass.
  3. C
    The lead sphere will experience a greater rate of acceleration from the exact moment of release because of its larger mass.
  4. D
    The aluminum sphere will take more time to reach its terminal velocity because its lower weight cannot overcome air resistance as quickly.

Answer

The lead sphere will attain a higher terminal velocity because more air resistance is required to counteract its greater weight.
An object reaches terminal velocity when the upward force of air resistance exactly equals its downward weight. Since the lead sphere has a much greater mass than the aluminum sphere, it has a proportionally greater weight. Because both spheres have identical dimensions and surface characteristics, they experience the exact same air resistance at any given speed. Therefore, the lead sphere must accelerate to a much higher velocity to generate enough air resistance to balance its larger weight.

Step-by-Step Solution

1
Analyze the forces acting on the spheres at the exact moment of release.
At exactly t=0t=0, velocity is zero, so aerodynamic drag is zero. Both spheres accelerate downward at exactly standard gravity (gg).
Gravity accelerates all objects equally in the absence of air resistance.
2
Determine how air resistance affects each sphere as they gain speed.
Because they are identical in size and shape, both spheres experience the exact same upward drag force at any given descent speed.
Aerodynamic drag depends on velocity, cross-sectional area, and shape, but not on the mass of the object.
3
Identify the conditions required to reach terminal velocity.
Terminal velocity is reached when the upward drag force exactly equals the downward weight.
When the net force is zero, acceleration stops and velocity becomes constant.
4
Compare the terminal velocities of the two spheres.
The lead sphere has a much larger mass, and therefore a larger weight. It must fall at a significantly faster speed to generate enough drag to equal its weight.
A larger downward force (weight) requires a proportionally larger upward force (drag) to achieve equilibrium.

Key Concept

Terminal Velocity and Atmospheric Drag
Estimated Time:1m 30s
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Topics

Gravity and Free Fall

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