The question asks for the force (in Newtons) that a trampoline must apply to a 49.0 kg gymnast to accelerate her straight up at $7.10 \, \text{m/s}^2$.

To solve this, we use Newton's second law of motion: $F_{\text{net}} = ma$ where:

• $F_{\text{net}}$ is the net force,
• $m$ is the mass of the gymnast, and
• $a$ is the acceleration.

In this scenario, the trampoline has to apply a force that overcomes both the force due to gravity and provide the additional upward acceleration.

The total force required from the trampoline is given by: $F_{\text{trampoline}} = m(g + a)$ where:

• $g = 9.81 \, \text{m/s}^2$ (acceleration due to gravity),
• $a = 7.10 \, \text{m/s}^2$ (given acceleration),
• $m = 49.0 \, \text{kg}$ (mass of the gymnast).

Calculation:

$F_{\text{trampoline}} = 49.0 \times (9.81 + 7.10)$ $F_{\text{trampoline}} = 49.0 \times 16.91$ $F_{\text{trampoline}} = 828.59 \, \text{N}$

Thus, the trampoline must apply a force of approximately 828.6 N.

Would you like more details or have any other questions?

Here are some related questions:

1. What is the net force if the gymnast is only in free fall (without the trampoline)?
2. How would the force change if the gymnast's mass was increased?
3. How does the direction of the gymnast's velocity affect the problem, if at all?
4. Can we neglect air resistance in this calculation? Why or why not?
5. How would the force applied by the trampoline change if the gymnast was accelerating downward instead of upward?

Tip: In problems like these, always make sure to account for both the force needed to overcome gravity and any additional force needed for acceleration.