A well-thrown ball is caught in a well-padded mitt. If the deceleration of the ball is\({\bf{2}}{\bf{.10 \times 1}}{{\bf{0}}^{\bf{4}}}\;{\bf{m/}}{{\bf{s}}^{\bf{2}}}\), and 1.85 ms (\({\bf{1}}\;{\bf{ms = 1}}{{\bf{0}}^{{\bf{ - 3}}}}\;{\bf{s}}\)) elapses from the time the ball first touches the mitt until it stops, what was the initial velocity of the ball?

The final velocity of ball is\(v = 0\)

The time for deceleration is\(t = 1.85\;{\rm{ms}} = 1.85 \times {10^{ - 3}}\;{\rm{s}}\)

The deceleration of ball is\(a = 2.10 \times {10^4}\;{\rm{m}}/{{\rm{s}}^2}\).

The initial velocity of the ball is found by using the first equation of motion and by considering the final velocity of ball as zero.

The initial velocity of ball is given as

\(v = u - at\)

Here, \(a\) is the deceleration of ball.

Substitute all the values in the above equation.

\(\begin{array}{l}0 = u - \left( {2.10 \times {{10}^4}\;{\rm{m}}/{{\rm{s}}^2}} \right)\left( {1.85 \times {{10}^{ - 3}}\;{\rm{s}}} \right)\\u = 38.85\;{\rm{m}}/{\rm{s}}\end{array}\)

Therefore, the initial velocity of ball is \(38.85\;{\rm{m}}/{\rm{s}}\).