A child, starting from rest at the top of a playground slide, reaches a speed of 7.0 meters per second at the bottom of the slide. what is the vertical height of the slide? [neglect friction.]

We know that the child will follow a accelerated movement due the gravity, according these equations:

h = v0 * t + 1/2 * g * t^2

v = v0 + g*t

where h is height, v is final speed, v0 is initial speed, g is gravity constant equal to 9.8 m/s2 and t is time

So the time to reach the bottom of the slide

v = v0 + g*t = > 7 = 0 + 9.8*t

t = 7/9.8 = 0.714 seconds

And then the height is:

h = v0*t + 1/2*g*t^2

h = 0*0.714 + 1/2*9.8*0.714^2 = 2.5 meters

So the height of the slide is 2.5 meters

for child sliding down;

at h the child’s potential energy: PE = m*g*h

at the bottom of the slide the kinetic energy: KE = (1/2) * m*V^2

m * g * h = (1/2) * m * v^2

g * h = (1/2) * v^2

h = (1/2) * (v^2) / g

h = (1/2) * (7^2) / 9.8

h = 2.5 meters