To calculate the magnitude of the acceleration of a skydiver when she is falling at half her terminal speed, we need to analyze the forces involved. The two primary forces acting on a skydiver during free fall are:

1. Gravitational Force (Weight): $F_g = mg$
2. Drag Force (Air Resistance): $F_d = \frac{1}{2} C_d \rho A v^2$, where:
   • $C_d$ is the drag coefficient,
   • $\rho$ is the air density,
   • $A$ is the cross-sectional area,
   • $v$ is the velocity of the skydiver.

At terminal velocity, the gravitational force is balanced by the drag force:

$mg = \frac{1}{2} C_d \rho A v_T^2$

Where $v_T$ is the terminal velocity. When the skydiver is falling at half her terminal velocity, the drag force will not fully counteract gravity.

Step-by-step solution:

1. At terminal velocity: The net force is zero because the gravitational force is balanced by the drag force: $F_g = F_d \quad \text{or} \quad mg = \frac{1}{2} C_d \rho A v_T^2$

2. At half terminal velocity: The drag force is proportional to the square of the velocity. Therefore, when the skydiver is falling at $\frac{v_T}{2}$, the drag force becomes: $F_d = \frac{1}{2} C_d \rho A \left( \frac{v_T}{2} \right)^2 = \frac{1}{4} \left( \frac{1}{2} C_d \rho A v_T^2 \right) = \frac{1}{4} mg$

   The drag force at half the terminal speed is one-fourth of the gravitational force.

3. Net force at half terminal speed: The net force acting on the skydiver is the difference between the gravitational force and the drag force: $F_{\text{net}} = mg - \frac{1}{4} mg = \frac{3}{4} mg$

4. Acceleration at half terminal speed: Using Newton's second law $F = ma$, the acceleration is: $a = \frac{F_{\text{net}}}{m} = \frac{\frac{3}{4} mg}{m} = \frac{3}{4} g$

Thus, the magnitude of the acceleration of the skydiver at the instant she is falling at one-half her terminal speed is:

$a = \frac{3}{4} g$

Final Answer:

The acceleration is $\frac{3}{4} \times 9.8 \, \text{m/s}^2 = 7.35 \, \text{m/s}^2$.

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Would you like any further details or clarifications on this solution?

Here are five related questions you might find interesting:

1. What factors influence the terminal velocity of a skydiver?
2. How does the drag coefficient $C_d$ vary with the shape and orientation of the skydiver?
3. What is the terminal velocity for a typical skydiver in different positions (e.g., belly-to-earth, head-down)?
4. How does air density impact the terminal velocity in high-altitude jumps?
5. What happens to a skydiver's acceleration as they approach terminal velocity?

Tip: In real-world scenarios, terminal velocity depends on body posture, clothing, and the altitude, since air density decreases with altitude!