Determine the velocity of the child just as it hits the floor:

v(i) = 0

v(f) = ??

d = 0.37 m

a = 9.8 \(\displaystyle\frac{{m}}{{s}^{{2}}}\)

\(\displaystyle{v}{\left({f}\right)}^{{2}}={v}{\left({i}\right)}^{{2}}+{2}{a}{d}\)

\(\displaystyle{v}{\left({f}\right)}^{{2}}={0}+{\left({2}\right)}{\left({9.8}\right)}{\left({0.37}\right)}\to\) initial velocity iszero

v(f) = 2.7 m/s

Now, determine the acceleration upon impact with the floor under both hardwood and carpet, knowing that the final velocity will be zero:

Hardwood:

\(\displaystyle{v}{\left({f}\right)}^{{2}}={v}{\left({i}\right)}^{{2}}+{2}{a}{d}\)

\(\displaystyle{0}={2.7}^{{2}}+{\left({2}\right)}{\left({a}\right)}{\left({0.0018}\right)}\to\) 1.8mm =.0018m

\(\displaystyle{a}=-\frac{{2.7}^{{2}}}{{0.0036}}\)

\(\displaystyle{a}=-{2025}\frac{{m}}{{s}^{{2}}}=-{2}\times{10}^{{3}}\frac{{m}}{{s}^{{2}}}\) (negative acceleration for slowing)

Carpet:

\(\displaystyle{v}{\left({f}\right)}^{{2}}={v}{\left({i}\right)}^{{2}}+{2}{a}{d}\)

\(\displaystyle{0}={2.7}^{{2}}+{\left({2}\right)}{\left({a}\right)}{\left({0.01}\right)}\to\) 1cm =.01m

\(\displaystyle{a}=-\frac{{2.7}^{{2}}}{{0.02}}\)

\(\displaystyle{a}=-{364.5}\frac{{m}}{{s}^{{2}}}=-{3.6}\times{10}^{{1}}\frac{{m}}{{s}^{{2}}}\) (negative acceleration for slowing)

v(i) = 0

v(f) = ??

d = 0.37 m

a = 9.8 \(\displaystyle\frac{{m}}{{s}^{{2}}}\)

\(\displaystyle{v}{\left({f}\right)}^{{2}}={v}{\left({i}\right)}^{{2}}+{2}{a}{d}\)

\(\displaystyle{v}{\left({f}\right)}^{{2}}={0}+{\left({2}\right)}{\left({9.8}\right)}{\left({0.37}\right)}\to\) initial velocity iszero

v(f) = 2.7 m/s

Now, determine the acceleration upon impact with the floor under both hardwood and carpet, knowing that the final velocity will be zero:

Hardwood:

\(\displaystyle{v}{\left({f}\right)}^{{2}}={v}{\left({i}\right)}^{{2}}+{2}{a}{d}\)

\(\displaystyle{0}={2.7}^{{2}}+{\left({2}\right)}{\left({a}\right)}{\left({0.0018}\right)}\to\) 1.8mm =.0018m

\(\displaystyle{a}=-\frac{{2.7}^{{2}}}{{0.0036}}\)

\(\displaystyle{a}=-{2025}\frac{{m}}{{s}^{{2}}}=-{2}\times{10}^{{3}}\frac{{m}}{{s}^{{2}}}\) (negative acceleration for slowing)

Carpet:

\(\displaystyle{v}{\left({f}\right)}^{{2}}={v}{\left({i}\right)}^{{2}}+{2}{a}{d}\)

\(\displaystyle{0}={2.7}^{{2}}+{\left({2}\right)}{\left({a}\right)}{\left({0.01}\right)}\to\) 1cm =.01m

\(\displaystyle{a}=-\frac{{2.7}^{{2}}}{{0.02}}\)

\(\displaystyle{a}=-{364.5}\frac{{m}}{{s}^{{2}}}=-{3.6}\times{10}^{{1}}\frac{{m}}{{s}^{{2}}}\) (negative acceleration for slowing)