Question

One 110-kg football lineman is running to the right at 2.75 m/s while another 125-kg lineman is running directly toward him at 2.60 m/s. What are (a) the magnitude and direction of the net momentum of these two athletes, and (b) their total kinetic energy?

Short answer

(a) 22.5 kg·m/s to the left, (b) 839.38 J.

Step-by-step solution

Calculate the Momentum of Each Lineman

Momentum is calculated using the formula $p=mv$, where $m$ is the mass, and $v$ is the velocity.- For the first lineman: $p_1=110\times 2.75=302.5\text{kg}\cdot \text{m/s}$.- For the second lineman: $p_2=125\times (-2.60)=-325\text{kg}\cdot \text{m/s}$ (the negative sign indicates opposite direction).

Determine the Net Momentum

To find the net momentum, sum the momentum of the two linemen:$$p_{net}=p_1+p_2=302.5-325=-22.5\text{kg}\cdot \text{m/s}$$The magnitude of the net momentum is $|-22.5|=22.5\text{kg}\cdot \text{m/s}$ and since it is negative, its direction is the same as the second lineman (to the left).

Calculate the Kinetic Energy of Each Lineman

Kinetic energy is calculated using the formula $KE=\frac{1}{2}m{v}^2$.- For the first lineman: $K{E}_1=\frac{1}{2}\times 110\times (2.75{)}^2=416.875\text{J}$.- For the second lineman: $K{E}_2=\frac{1}{2}\times 125\times (2.60{)}^2=422.5\text{J}$.

Determine the Total Kinetic Energy

Sum the kinetic energy of both linemen:$$K{E}_{total}=K{E}_1+K{E}_2=416.875+422.5=839.375\text{J}$$

Key concepts

Momentum

Momentum is a key concept in physics that describes the quantity of motion an object possesses. It is given by the product of an object's mass and its velocity: $p=mv$. This makes momentum a vector quantity, meaning it has both magnitude and direction. When calculating momentum, you must consider the direction of movement—positive for one direction and negative for the opposite. In the case of the two linemen coliding, each has their own momentum based on their mass and speed. With the first lineman moving to the right and the second moving to the left, their momenta will have opposite signs. Being able to compute momentum accurately helps predict how these forces interact during collisions.

Kinetic Energy

Kinetic Energy is the energy an object has due to its motion. It's calculated using the formula $KE=\frac{1}{2}m{v}^2$, where $m$ is the mass, and $v$ is the velocity of the object.Understanding kinetic energy is important because it quantifies the amount of work an object can do due to its motion. In the example of the two linemen, each has energy based on their speed and mass. The faster and heavier a person (or object) is, the more kinetic energy they possess. Summing up their kinetic energies gives the total energy available just before they collide.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, with no external forces, the total momentum before and after an event remains constant. This is crucial in collision physics, where understanding momentum helps determine the results of collisions. In the scenario of the colliding linemen, despite their individual momentum values being different, their combined momentum can help predict the outcome post-collision. Even though they are moving towards each other, their net momentum indicates the direction the result of the collision will lead.

Collision Physics

Collision physics examines how objects interact when they crash into each other. Two key factors govern these interactions:

• Momentum: Dictates the direction and magnitude of the resulting motion.
• Kinetic Energy: Determines the potential damage or energy transfer during the impact.

In perfectly elastic collisions, both momentum and kinetic energy are conserved. But real-world collisions often involve energy transformations, meaning kinetic energy is not fully conserved due to factors like friction or deformation. Understanding these interactions helps in analyzing how the two linemen, with given masses and velocities, interact during their eventual meeting.