Question

Attempting to score a touchdown, an \(85.0-\mathrm{kg}\) tailback jumps over his blockers, achieving a horizontal speed of \(8.90 \mathrm{~m} / \mathrm{s}\). He is met in midair just short of the goal line by a \(110 .-\mathrm{kg}\) linebacker traveling in the opposite direction at a speed of \(8.00 \mathrm{~m} / \mathrm{s}\). 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

Answer: The final speed of the entangled players is approximately -1.18 m/s, and the tailback does not score a touchdown.

Step-by-step solution

Analyze the given data and understand the problem

We are given the mass and horizontal speed of both players before the collision. Our task is to find the final speed of the entangled players and determine whether the tailback scores a touchdown.

Use the conservation of momentum

According to the conservation of momentum principle, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be written as: \(m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f\) where: - \(m_1\) is the mass of the tailback (\(85.0\,\text{kg}\)) - \(v_1\) is the horizontal speed of the tailback (\(8.90\,\text{m/s}\)) - \(m_2\) is the mass of the linebacker (\(110\,\text{kg}\)) - \(v_2\) is the horizontal speed of the linebacker (since they are going in the opposite direction, we can write their speed as \(-8.00\,\text{m/s}\)) - \(v_f\) is the final speed of the entangled tailback and linebacker

Solve for the final speed\(v_f\)

Plug the given values into the momentum conservation equation and solve for \(v_f\): \(85.0\,\text{kg} \cdot 8.90\,\text{m/s} + 110\,\text{kg} \cdot (-8.00\,\text{m/s}) = (85.0\,\text{kg} + 110\,\text{kg})v_f\) Now, solve for \(v_f\): \(v_f = \dfrac{85.0\,\text{kg} \cdot 8.90\,\text{m/s} - 110\,\text{kg} \cdot 8.00\,\text{m/s}}{85.0\,\text{kg} + 110\,\text{kg}} \approx -1.18\,\text{m/s}\) The negative sign indicates that the final direction of motion is opposite to that of the tailback's initial direction.

Determine if the tailback scores a touchdown

Since the final speed is negative, the entangled players will move in the opposite direction of the tailback's initial direction (toward the linebacker's initial direction) upon collision. Therefore, the tailback will not score a touchdown in this scenario. In summary, the final speed of the entangled players is approximately \(-1.18\,\text{m/s}\), and the tailback will not score a touchdown.

Key concepts

Momentum Collision Problem

When two objects collide, be it football players or billiard balls, we witness a 'momentum collision problem.' Such a problem involves calculating the resulting speed and direction of objects after they collide.

In our textbook example, we have two players, a tailback and a linebacker, who collide on the field. By applying physics principles, particularly the conservation of momentum, we can predict the aftermath of this high-impact sports collision. The scenario requires an understanding of mass, velocity, and the fundamental tenet that in an isolated system, total momentum remains constant, regardless of the interaction between the objects.

To solve these problems, we must gather the mass and velocity of each object before the collision, identify the direction of motion (choosing a frame of reference like treating one direction as positive and the opposite as negative), and then apply the momentum conservation equation to find out the final velocity of the combined mass post-collision.

Physics of Collisions

In the 'physics of collisions,' we differentiate between elastic and inelastic collisions. An elastic collision is one where both momentum and kinetic energy are conserved. Objects bounce off each other with no loss in their overall kinetic energy. However, an inelastic collision, which is the case with our football players, is where objects collide and move together with a common velocity after impact, resulting in a loss of kinetic energy.

An understanding of collision types is crucial in assessing the end results. For instance, in our problem about the football players, after the impact, they move together, indicating an inelastic collision. This is important for predicting whether the tailback scores a touchdown. Because the total system kinetic energy is not conserved, some of it is converted into other forms of energy, such as sound, heat, or even the deformation of the equipment they wear.

Momentum Conservation Equation

The 'momentum conservation equation' is pivotal in resolving problems involving collisions:
\[(m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f)\]
Here, \(m_1\) and \(m_2\) represent the masses of the colliding objects, \(v_1\) and \(v_2\) are their velocities before collision, and \(v_f\) is the final velocity after collision. It's a vector equation, meaning it takes into account the direction of motion.

In the step-by-step solution, we see the equation being applied to solve for the final speed. It illustrates that when we know the initial speeds and masses, we can confidently determine the outcome post-collision. For effective learning, students should practice applying the conservation of momentum equation to different scenarios to reinforce the concept.