Let (g, ∗) be a set with a composition law. an element e ∈ g is called neutral if the equalities a ∗ e = a and e ∗ a = a both hold for all a ∈ g. (a) prove that there can be at most one neutral element e. (b) give an example of a pair (g, ∗) which does not admit a neutral element.

7.9 m/s

Explanation:

A right triangle is formed where one leg (let's call it x) form an angle of 47° with the hypotenuse and the hypotenuse is the rope. From cosine definition:

cos 47° = x/10

x = 10*cos 47°

x = 6.82 m

The difference between the rope length and this value gives us the height of the child before her father releases the swing. Let's call this height h.

h = 10 m - 6.82 m = 3.18 m

Before the swing starts the child is not moving, then all of her energy is potential energy (PE). When she gets at the bottom of the arc of the swing,  all of her PE is transformed into kinetic energy (KE).

KE = PE

(1/2)*m*v² = m*g*h

where v is her speed and g is the standard gravitational acceleration. Solving for v:

v = √(2*g*h)

v = √(2*9.81*3.18)

v = 7.9 m/s