Answer :
To determine their mutual speed immediately after the collision, we can employ the principle of conservation of momentum. In a collision where no external forces are acting, the total momentum before the collision is equal to the total momentum after the collision.
Step-by-Step Solution:
Define the Variables:
- Mass of the tackler, [tex]m_1 = 140 \text{ kg}[/tex]
- Initial velocity of the tackler, [tex]v_1 = 2.1 \text{ m/s}[/tex]
- Mass of the halfback, [tex]m_2 = 93 \text{ kg}[/tex]
- Initial velocity of the halfback, [tex]v_2 = 4.9 \text{ m/s}[/tex]
Set Up the Equation for Conservation of Momentum:
- Total initial momentum = Total final momentum
- [tex]m_1v_1 + m_2v_2 = (m_1 + m_2) v_f[/tex]
- Here, [tex]v_f[/tex] is the final velocity they have together after collision.
Calculate the Initial Momentums:
- Momentum of the tackler: [tex]140 \times 2.1 = 294 \text{ kg m/s}[/tex]
- Since the tackler and the halfback are moving in opposite directions, we assign a negative sign to the halfback's velocity:
- Momentum of the halfback: [tex]93 \times (-4.9) = -455.7 \text{ kg m/s}[/tex]
Calculate the Total Initial Momentum:
- [tex]294 - 455.7 = -161.7 \text{ kg m/s}[/tex]
Plug into the Conservation of Momentum Equation:
- [tex]-161.7 = (140 + 93) v_f[/tex]
- [tex]-161.7 = 233 v_f[/tex]
Solve for the Final Velocity [tex]v_f[/tex]:
- [tex]v_f = \frac{-161.7}{233} \approx -0.694 \text{ m/s}[/tex]
Conclusion:
The negative sign indicates that after the collision, both players move in the direction the halfback was initially heading (the negative direction we chose to signify his movement). The mutual speed immediately after the collision is approximately [tex]-0.69 \text{ m/s}[/tex]. The mutual speed expressed using two significant figures is [tex]-0.69 \text{ m/s}[/tex].
