Final answer:
To accelerate a 49-kg acrobat straight up at 7.9 m/s², the trampoline must apply a total force of approximately 867.3 newtons, calculated using the formula F = m(a + g), where g is the acceleration due to gravity (9.8 m/s²).
Step-by-step explanation:
To determine the force the trampoline must apply to accelerate a 49-kg acrobat straight up at 7.9 m/s2, we use 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 its acceleration (F = ma). In this scenario, the trampoline must not only apply a force to provide the upward acceleration but also counteract the force of gravity acting on the acrobat (her weight).
The force of gravity (weight) is given by W = mg, where g is the acceleration due to gravity (9.8 m/s2). The total force the trampoline must apply is the sum of the force needed for upward acceleration and the force to support the acrobat's weight. Therefore, the total force (F) is calculated as follows:
F = ma + mg
F = m(a + g)
Plugging in the values:
F = 49 kg (7.9 m/s2 + 9.8 m/s2)
F = 49 kg (17.7 m/s2)
F = 867.3 N
Thus, the trampoline must apply a force of approximately 867.3 newtons to accelerate the acrobat straight up at 7.9 m/s2.