Question

A 95-kg fullback is running at \({\bf{3}}{\bf{.0}}\;{{\bf{m}} \mathord{\left/{\vphantom {{\bf{m}} {\bf{s}}}} \right.\\} {\bf{s}}}\) to the east and is stopped in 0.85 s by a head-on tackle by a tackler running due west. Calculate

(a) the original momentum of the fullback,

(b) the impulse exerted on the fullback,

(c) the impulse exerted on the tackler, and

(d) the average force exerted on the tackler.

Short answer

(a) The original momentum of the fullback is \(285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\) towards the east.

(b) The impulse exerted on the fullback is \( - 285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\) towards the west.

(c) The impulse exerted on the tackler is \(285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\).

(d) The average force exerted on the tackler is \(335.3\;{\rm{N}}\).

Step-by-step solution

Define impulse

The impulse exerted on an object is the product of the applied force and the elapsed time. It varies directly with the applied force.

Given information

The mass of the fullback is \({m_1} = 95\;{\rm{kg}}\).

The initial speed of the fullback is \({\vec v_{1,{\rm{i}}}} = 3.0\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.\\} {\rm{s}}}\).

The time interval is \(\Delta {t_1} = 0.85\;{\rm{s}}\).

Since at the final position, the fullback is stopped, his final velocity is \({\vec v_{1,{\rm{f}}}} = 0\).

Calculate the original momentum of the fullback

(a)The original momentum of the fullback can be calculated as shown below:

\(\begin{array}{c}{{\vec P}_1} = {m_1}{{\vec v}_{1,{\rm{i}}}}\\{{\vec P}_1} = \left( {95\;{\rm{kg}}} \right)\left( {3.0\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.\\} {\rm{s}}}} \right)\\{{\vec P}_1} = 285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\end{array}\)

Thus, the original momentum of the fullback is \(285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\) towards the east.

Calculate the impulse exerted on the fullback

(b)

The impulse exerted on the fullback can be calculated as shown below:

\(\begin{array}{c}{{\vec F}_1}\Delta t = {m_1}\left( {{{\vec v}_{1,{\rm{f}}}} - {{\vec v}_{1,{\rm{i}}}}} \right)\\{{\vec F}_1}\Delta t = \left( {95\;{\rm{kg}}} \right)\left[ {\left( 0 \right) - \left( {3.0\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.\\} {\rm{s}}}} \right)} \right]\\{{\vec F}_1}\Delta t = - 285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\end{array}\)

Thus, the impulse exerted on the fullback is \( - 285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\) towards the west.

Calculate the impulse exerted on the tackler

(c)The impulse exerted on the tackler is equal in magnitude to that exerted on the fullback but in the opposite direction. That is, \({\vec F_2}\Delta t = 285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\).

Thus, the impulse exerted on the tackler is \(285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}\).

Calculate the average force exerted on the tackler

(d)The average force exerted on the tackler can be calculated as shown below:

\(\begin{array}{c}{{\vec F}_2} = \frac{{{m_2}\left( {{v_{2,{\rm{f}}}} - {v_{2,{\rm{i}}}}} \right)}}{{\Delta {t_1}}}\\{{\vec F}_2} = \frac{{{{\vec F}_2}\Delta t}}{{\Delta {t_1}}}\\{{\vec F}_2} = \frac{{\left( {285\;{{{\rm{kg}} \cdot {\rm{m}}} \mathord{\left/{\vphantom {{{\rm{kg}} \cdot {\rm{m}}} {\rm{s}}}} \right.\\} {\rm{s}}}} \right)}}{{\left( {0.85\;{\rm{s}}} \right)}}\\{{\vec F}_2} = 335.3\;{\rm{N}}\end{array}\)

Thus, the average force exerted on the tackler is \(335.3\;{\rm{N}}\).