Problem 73 - University Physics with Modern Physics

Question

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)?

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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

Step 3: 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

Step 4: 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

Step 5: 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.

Key Concepts

Conservation of Momentum

A fundamental principle in physics, conservation of momentum states that in a closed system, the momentum before any interaction is equal to the momentum after the interaction.
- Momentum is a vector quantity, meaning it has both magnitude and direction. - It is calculated as the product of an object's mass (\( m \)) and its velocity (\( v \)).
When two players collide in a football game, like the tailback and the linebacker in our exercise, we apply this principle because no external forces significantly impact their momentum during the brief instant they collide.
In our scenario, both players' initial momentum must be considered. The tailback moves forward with a given momentum, and the linebacker moves backwards with an opposing momentum.
The total momentum of the system is calculated by adding these two momentum values, considering their respective directions by assigning a negative value to one (in our case, the linebacker).
The law dictates that after they collide and stick together (an inelastic collision), their combined momentum remains the same as the total initial momentum.
Thus, conservation of momentum ensures that even if their velocities change, the total product of mass and velocity remains constant.

Collisions in Physics

Collisions in physics can be categorized primarily into elastic and inelastic types.
- Elastic collisions occur when the total kinetic energy of the system is conserved. - Inelastic collisions, however, do not conserve kinetic energy. Instead, they often result in the objects sticking together or deforming.
Football challenges like the one between the 85-kg tailback and the 110-kg linebacker are typically inelastic because the players often remain entangled after the impact.
In an inelastic collision, the post-collision system behaves as a single mass.
In our situation, after the tailback and linebacker collide and grab hold of each other, they're considered as one moving mass. This simplifies calculations, allowing us to find the new single velocity using the combined mass and total momentum of the system.
By understanding these basics of collisions in physics, you can better grasp how athletes' movements and interactions work in sports governed by these principles.

Kinematics in Sports

Kinematics involves studying the motion of objects without considering the forces causing the motion. It helps us understand athletes' movements, providing insights into efficiency and effectiveness in sports.
In our example, the motion of the tailback and linebacker is analyzed to determine their final velocity after impacting.
- The horizontal speed they maintain or change is crucial, particularly in sports like football, where reaching a touchdown is the main objective of the tailback.
By integrating concepts like velocity and time with the principles of the momentum conservation discussed earlier, kinematics allows us to predict the result of such physical collisions.
In essence, it evaluates how movements transition, helping to chart paths taken by athletes during competition.
With kinematics, players can refine their runs, jumps, and tackles, adjusting their strategies based on their movements' effectiveness—empowering them to succeed in achieving their sporting goals.