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Chapter 7: Problem 73

Attempting to score a touchdown, an 85-kg tailback jumps over his blockers, achieving a horizontal speed of \(8.9 \mathrm{~m} / \mathrm{s} .\) He is met in midair just short of the goal line by a 110 -kg linebacker traveling in the opposite direction at a speed of \(8.0 \mathrm{~m} / \mathrm{s}\). The linebacker grabs the tailback. a) What is the speed of the entangled tailback and linebacker just after the collision? b) Will the tailback score a touchdown (provided that no other player has a chance to get involved, of course)?

Short Answer

Expert verified
Answer in 2-3 sentences, using the given information and calculations.

Step by step solution

01

Calculate initial momentum

Calculate the initial momentum of each player. Momentum (p) is defined as the mass (m) times the velocity (v): p = m * v. For the tailback: p_tb = m_tb * v_tb For the linebacker: p_lb = m_lb * v_lb And, m_tb = 85 kg, v_tb = 8.9 m/s (going forward) m_lb = 110 kg, v_lb = 8.0 m/s (going backward) Note: As both players are moving in opposite directions, we should take the velocity of one of them as negative. In this case, let's consider the velocity of the linebacker negative.
02

Calculate total initial momentum

Add the initial momentum of each player to find the total initial momentum of the system. p_total_initial = p_tb + p_lb
03

Calculate final momentum

By the law of conservation of momentum, the final momentum of the system is equal to the total initial momentum. Therefore, p_total_final = p_total_initial
04

Calculate final velocity of the entangled system

As both the tailback and linebacker are entangled, their total mass can be considered as a single entity. Let's denote the final velocity of the entangled system by v_final. m_total = m_tb + m_lb Then, by the momentum formula, we have: p_total_final = m_total * v_final Now we can solve for v_final: v_final = p_total_final / m_total
05

Calculate if the tailback scores a touchdown

After calculating v_final, if it is positive, that means the tailback (and the entangled system) is still moving forward and has a chance to score a touchdown. If v_final is negative or zero, then the tailback (and system) stopped or moved backward, and the touchdown was not scored. Now, we can perform the calculations and determine if the tailback scores a touchdown.

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