Chapter 9, Problem 96A

(a)

To determine

To sketch:The situations before and after collision.

Answer to Problem 96A

The sketch of the situations before and after collision is shown in Figure 1.

Explanation of Solution

Given info:

Consider the mass of fullback as $m_{\text{A}}=92\text{kg}$ and the mass of line backers are $m_{\text{B}}=m_{\text{C}}=75\text{ kg}$ .

Consider the initial velocity towards a goal line for a touchdown is $v_{\text{Ai}}=5\text{m}/\text{s}$ .

The initial velocity of one line backer $v_{\text{Bi}}=-2.0\text{m}/\text{s}$ .

The initial velocity of other line backer $v_{\text{Ci}}=-4.0\text{m}/\text{s}$

Calculation:

For the given specifications, the situation before and after collision are shown below.

  

Figure 1

A full back is running towards the positive x-direction, with a velocity $v_{\text{Ai}}$ while two line-backers are running towards negative x-direction, with a velocity $v_{\text{Bi}}$ and $v_{\text{Ci}}$ respectively.

Conclusion:

Therefore, the sketch of the situations before and after collision is shown in the figure 1.

(b)

To determine

The velocity of the players after collision.

Answer to Problem 96A

The velocity of the players after collision is $v_{\text{f}}=0.041\text{m}/\text{s}$ .

Explanation of Solution

Given:

Consider the mass of fullback as $m_{\text{A}}=92\text{kg}$ and the mass of line backers are $m_{\text{B}}=m_{\text{C}}=75\text{ kg}$ .

Consider the initial velocity towards a goal line for a touchdown is $v_{\text{Ai}}=5\text{m}/\text{s}$ .

The initial velocity of one line backer $v_{\text{Bi}}=-2.0\text{m}/\text{s}$ .

The initial velocity of other line backer $v_{\text{Ci}}=-4.0\text{m}/\text{s}$

Formula used:

The expression for the momentum is,

  $p=mv$

Here, $m$ is the mass of the system and $v$ is the velocity of the system.

Calculation:

Consider the final velocity of the fullback and the line bakers are $v_{\text{Af}}=v_{\text{Bf}}=v_{\text{Cf}}=v_{\text{f}}$ .

Refer to Figure 1 given in part (a).

By the law of conservation of momentum, the initial momentum of the system is equal to the final momentum of the system.

  $\begin{array}{l}{p}_{\text{i}}=p_{\text{f}}\\ p_{\text{Ai}}+p_{\text{Bi}}+p_{\text{Ci}}=p_{\text{Af}}+p_{\text{Bf}}+p_{\text{Cf}}\\ m_{\text{A}}{v}_{\text{Ai}}+m_{\text{B}}{v}_{\text{Bi}}+m_{\text{C}}{v}_{\text{Ci}}=m_{\text{A}}{v}_{\text{Af}}+m_{\text{B}}{v}_{\text{Bf}}+m_{\text{C}}{v}_{\text{Cf}}\\ m_{\text{A}}{v}_{\text{Ai}}+m_{\text{B}}{v}_{\text{Bi}}+m_{\text{C}}{v}_{\text{Ci}}=\left(m_{\text{A}}+m_{\text{B}}+m_{\text{C}}\right)v_{\text{f}}\end{array}$

  $\begin{array}{l}\frac{m_{\text{A}}{v}_{\text{Ai}}+m_{\text{B}}{v}_{\text{Bi}}+m_{\text{C}}{v}_{\text{Ci}}}{\left(m_{\text{A}}+m_{\text{B}}+m_{\text{C}}\right)}=v_{\text{f}}\\ v_{\text{f}}=\frac{\left(92\text{ kg}\times 5\text{m}/\text{s}\right)+\left(\text{75 kg}\times \left(-2\text{m}/\text{s}\right)\right)+\left(\text{75 kg}\times \left(-4\text{m}/\text{s}\right)\right)}{\left(92\text{ kg}\right)+\left(\text{75 kg}\right)+\left(\text{75 kg}\right)}\\ v_{\text{f}}=\frac{10}{242}\\ v_{\text{f}}=0.041\text{m}/\text{s}\end{array}$

Conclusion:

Therefore, the velocity of the players after collision is $v_{\text{f}}=0.041\text{m}/\text{s}$ .

(c)

To determine

To identify:Whether the fullback scores a touchdown.

Answer to Problem 96A

Yes, the fullback scores a touchdown.

Explanation of Solution

Given:

Consider the mass of fullback as $m_{\text{A}}=92\text{kg}$ and the mass of line backers are $m_{\text{B}}=m_{\text{C}}=75\text{ kg}$ .

Consider the initial velocity towards a goal line for a touchdown is $v_{\text{Ai}}=5\text{m}/\text{s}$ .

The initial velocity of one line backer $v_{\text{Bi}}=-2.0\text{m}/\text{s}$ .

The initial velocity of other line baker $v_{\text{Ci}}=-4.0\text{m}/\text{s}$

Formula used:

The expression for the momentum is,

  $p=mv$

Here, $m$ is the mass of the system and $v$ is the velocity of the system.

Calculation:

From part (b), the velocity of the players after collision is $v_{\text{f}}=0.041\text{m}/\text{s}$ .

The velocity of the players after collision is positive. This shows that the velocity of the players is in the direction of the fullback. So the fullback scores a touchdown.

Hence, the fullback can cross the goal line for a touchdown.

Conclusion:

Therefore, the fullback scores a touchdown.