Final answer:
The force a trampoline must apply to accelerate a 45.0-kg gymnast upwards at a rate of 7.50 m/s² is calculated using Newton's second law of motion. By considering both the force necessary for acceleration and the gymnast's weight, we find that the trampoline must provide an additional force of 337.5 N, which is option a).
Step-by-step explanation:
The question asks us to determine the force a trampoline must apply to accelerate a 45.0-kg gymnast upwards at a rate of 7.50 m/s². To find this force, we must consider Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by the acceleration it experiences (F = ma). Additionally, we must account for the force of gravity, which acts downward on the gymnast as weight (w = mg).
The total force exerted by the trampoline is the sum of the force required to provide the upward acceleration and the weight of the gymnast. Mathematically, this is reflected by the formula ΣF = ma + w, where ΣF is the net force, m is the mass, a is the acceleration, w is the weight, and g is the acceleration due to gravity (9.80 m/s²). Plugging in the values, we get F = 45.0 kg × 7.50 m/s² + 45.0 kg × 9.80 m/s², resulting in F = 337.5 N + 441 N = 778.5 N.
However, since the question asks for the additional force the trampoline must apply, aside from the weight that is already supporting the gymnast, we need to subtract the weight from the total force. So the actual force provided by the trampoline for the acceleration alone is F₀ = 778.5 N - 441 N = 337.5 N. Therefore, the correct option is a) 337.5 N.