Final answer:
The gymnast must exert a total force of 3139.2 N to decelerate at 7 times the acceleration due to gravity, which is not listed in the given options.
Step-by-step explanation:
To calculate the force exerted by a 40.0-kg gymnast when landing, we first determine the deceleration she experiences, which is given as 7.00 times the acceleration due to gravity (g = 9.81 m/s2). Thus, the deceleration is 7.00 × 9.81 m/s2.
The force (F) needed to cause this deceleration can be found using Newton's second law of motion, which is F = m × a, where m is the mass of the gymnast and a is the deceleration. So, F = 40.0 kg × (7.00 × 9.81 m/s2).
Upon calculation, F = 40.0 kg × 68.67 m/s2 = 2746.8 N. However, we must also consider the force of gravity acting on the gymnast, which is mg, where m is her mass. The total force she must exert must overcome both her weight and the force to decelerate, so the total force is F_total = mg + F. Hence, F_total = (40.0 kg × 9.81 m/s2) + 2746.8 N = 392.4 N + 2746.8 N.
After adding both forces, F_total = 3139.2 N. Therefore, the gymnast must exert a force equal to 3139.2 N to achieve the required deceleration.
Comparing this with the options given, option c) 700 N is incorrect, and none of the options (a) 280 N, (b) 490 N, (d) 980 N represent the calculated total force. There must be an error in the options provided as the correct force is 3139.2 N, which is not listed.