Question

A football quarterback runs 15.0 m straight down the playing field in 2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He breaks the tackle and runs straight forward another 21.0 m in 5.20 s. Calculate his average velocity (a) for each of the three intervals and (b) for the entire motion.

Short answer

(a) The average velocities for first-, second- and third-time intervals are\({\bf{6}}\;{\bf{m/s}}\),\({\bf{ - 1}}{\bf{.714}}\;{\bf{m/s}}\)and\({\bf{4}}{\bf{.04}}\;{\bf{m/s}}\).

(b) The average velocity for entire motion is \({\bf{3}}{\bf{.5}}\;{\bf{m/s}}\).

Step-by-step solution

Determination of average velocity for each time interval (a)

Given Data:

The displacement for first time interval is \({d_1} = 15\;{\rm{m}}\)

The duration for first time interval is\({t_1} = 2.50\;{\rm{s}}\)

The displacement for second time interval is \({d_2} = - 3\;{\rm{m}}\)

The duration for second time interval is\({t_2} = 1.75\;{\rm{s}}\)

The displacement for third time interval is \({d_3} = 21\;{\rm{m}}\)

The duration for third time interval is\({t_3} = 5.20\;{\rm{s}}\)

The average velocity of an object is the variation in the displacement with time. It is greater, less or equal to the average speed.

The average velocity for first time interval is given as

\(\begin{array}{l}{v_1} = \frac{{{d_1}}}{{{t_1}}}\\{v_1} = \frac{{15\;{\rm{m}}}}{{2.50\;{\rm{s}}}}\\{v_1} = 6\;{\rm{m}}/{\rm{s}}\end{array}\)

The average velocity for second time interval is given as

\(\begin{array}{l}{v_2} = \frac{{{d_2}}}{{{t_2}}}\\{v_2} = \frac{{ - 3\;{\rm{m}}}}{{1.75\;{\rm{s}}}}\\{v_2} = - 1.714\;{\rm{m}}/{\rm{s}}\end{array}\)

The average velocity for third time interval is given as

\(\begin{array}{l}{v_3} = \frac{{{d_3}}}{{{t_3}}}\\{v_3} = \frac{{21\;{\rm{m}}}}{{5.20\;{\rm{s}}}}\\{v_3} = 4.04\;{\rm{m}}/{\rm{s}}\end{array}\)

Therefore, the average velocities for first-, second- and third-time intervals are \(6\;{\rm{m}}/{\rm{s}}\), \( - 1.714\;{\rm{m}}/{\rm{s}}\) and \(4.04\;{\rm{m}}/{\rm{s}}\).

Determination of average velocity for entire motion (b)

The average velocity for entire motion is given as

\(v = \frac{{{d_1} + {d_2} + {d_3}}}{{{t_1} + {t_2} + {t_3}}}\)

Substitute all the values in the above equation.

\(\begin{array}{l}v = \frac{{15\;{\rm{m}} + \left( { - 3\;{\rm{m}}} \right) + 21\;{\rm{m}}}}{{2.50\;{\rm{s}} + 1.75\;{\rm{s}} + 5.20\;{\rm{s}}}}\\v = 3.5\;{\rm{m}}/{\rm{s}}\end{array}\)

Therefore, the average velocity for entire motion is \(3.5\;{\rm{m}}/{\rm{s}}\).