is someone can solve this accurately, I will be so happy?

Question

At a local playground, a child on a swing has a speed of 3.4 m/s when the swing is at its lowest point.

a.) To what maximum vertical height (above the bottom of the swing) does the child rise, assuming he sits still and "coasts"?
m

b.) Reanswer part a, with the initial speed of the child being half as much.
m

c.) During this process, how much work is done by the tension in the rope? (What assumptions did you make in part a?)
J

Answer 1

The question can be answered using conservation of energy

At the bottom of the swing the energy of the swinging child is kinetic

.5*m*v^2

as the child rises without any loss of energy in the system, the kinetic energy gets converted to potential energy
m*g*h

where h is the vertical height.

Using 9.81 as g

h=.5*v^2/g

For v=3.4
h=0.589 m

for v=3.4/2
h=0.147 m

The work done by the tension in the rope is zero since it is perpendicular to the displacement of the child and
Work=F*displacement

The assumptions I made in A is that the child is a point mass swinging like a rigid pendulum with no friction or air resistance.

j

Answer 2

To answer part (a) you take the KE and equate it to the gain in GPE.
1/2mv^2=mgh
h=v^2/2g

Taking g as 9.81 ms^-2 gives the height as 0.59m

(b) same process gives height as 0.15m

(c) Work done = energy transferred. Since no energy is transferred to or from the child then no work is done.

Of course this ignores the effects of friction and air resistance as well as assumes that the child 'coasts'.

The answer...
"The work done by the tension in the rope is zero since it is perpendicular to the displacement of the child and
Work=F*displacement"
has an incorrect reasoning.

Firstly tension is not the only force acting, there is also the child's weight to consider, which acts in the downward vertical direction.

Secondly, the direction of the tension changes as teh child gains height. At the lowest point tension is in the vertical direction, at the heighest point it acts at an angle.