Final answer:
To find the superhero's speed at the bottom of the swing, we can use conservation of mechanical energy. At the top of the swing, all of the gravitational potential energy is converted to kinetic energy, so we can equate the two.
Step-by-step explanation:
To find the superhero's speed at the bottom of the swing, we can use conservation of mechanical energy. At the top of the swing, all of the gravitational potential energy is converted to kinetic energy, so we can equate the two:
mg * h = (1/2) * m * v^2
where mg is the weight of the superhero (mass times acceleration due to gravity), h is the height of the swing, m is the mass of the superhero, and v is the speed at the bottom of the swing. Rearranging the equation, we can solve for v:
v = sqrt(2gh)
where sqrt means square root.
In part (a), the superhero starts from rest, so her initial speed is 0. Plugging in the values, we get:
v = sqrt(2 * 9.8 m/s^2 * 5.57 m) = 10.95 m/s
In part (b), the superhero pushes off with a speed of 5.40 m/s. The initial kinetic energy is given by (1/2) * m * v^2. Adding this to the gravitational potential energy, we can solve for v:
v = sqrt(2gh + v0^2)
where v0 is the initial speed. Plugging in the values, we get:
v = sqrt(2 * 9.8 m/s^2 * 5.57 m + (5.40 m/s)^2) = 16.87 m/s