A 91.0 kg fullback running east with a speed of 6.00 m/s is tackled by a 73.0 kg opponent running north with a speed of 3.00 m/s. Where did the lost energy go?
To determine where the lost energy went in this scenario, we need to consider the concept of conservation of momentum.
Momentum (p) is a vector quantity and is defined as the product of an object's mass (m) and its velocity (v). The law of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, as long as there is no external force acting on the system.
In this case, the fullback and the opponent collide. To find out where the lost energy goes, we need to calculate the velocities of the two players after the collision and compare them to their initial velocities.
The initial momentum of the fullback (m₁v₁) can be calculated as:
p₁ = m₁v₁ = (91.0 kg) * (6.00 m/s) = 546 kg·m/s
The initial momentum of the opponent (m₂v₂) can be calculated as:
p₂ = m₂v₂ = (73.0 kg) * (3.00 m/s) = 219 kg·m/s
The total initial momentum before the collision is:
p_initial = p₁ + p₂ = 546 kg·m/s + 219 kg·m/s = 765 kg·m/s
After the collision, the players will have new velocities. To calculate these velocities, we can use the equations derived from the conservation of momentum:
m₁v₁i + m₂v₂i = m₁v₁f + m₂v₂f
Where:
m₁ = mass of fullback
v₁i = initial velocity of fullback
m₂ = mass of opponent
v₂i = initial velocity of opponent
v₁f = final velocity of fullback
v₂f = final velocity of opponent
Now, let's substitute the given values into the equation and solve for the final velocities:
(91.0 kg) * (6.00 m/s) + (73.0 kg) * (3.00 m/s) = (91.0 kg) * v₁f + (73.0 kg) * v₂f
546 kg·m/s + 219 kg·m/s = (91.0 kg) * v₁f + (73.0 kg) * v₂f
765 kg·m/s = (91.0 kg) * v₁f + (73.0 kg) * v₂f
Now, since the question asks about the lost energy, we need to calculate the energy before and after the collision.
The initial kinetic energy of the system can be calculated as:
KE_initial = 0.5 * m₁ * (v₁i)² + 0.5 * m₂ * (v₂i)²
The final kinetic energy of the system can be calculated as:
KE_final = 0.5 * m₁ * (v₁f)² + 0.5 * m₂ * (v₂f)²
Substituting the given values and solving for both initial and final kinetic energies:
KE_initial = 0.5 * (91.0 kg) * (6.00 m/s)² + 0.5 * (73.0 kg) * (3.00 m/s)²
KE_initial = 1239 Joules
KE_final = 0.5 * (91.0 kg) * (v₁f)² + 0.5 * (73.0 kg) * (v₂f)²
Now, we can solve for the final velocities by rearranging the equation:
KE_final - KE_initial = 0
0.5 * (91.0 kg) * (v₁f)² + 0.5 * (73.0 kg) * (v₂f)² - 1239 Joules = 0
Substituting the known values and solving for the final velocities:
0.5 * (91.0 kg) * (v₁f)² + 0.5 * (73.0 kg) * (v₂f)² - 1239 Joules = 0
By solving this equation, we can find the final velocities of the fullback and the opponent after the collision.
When we solve for the final velocities, we can calculate the final kinetic energy using the equation:
KE_final = 0.5 * m₁ * (v₁f)² + 0.5 * m₂ * (v₂f)²
The difference between the initial and final kinetic energies will give us the lost energy, which can be attributed to various factors such as sound, heat, and deformation during the collision.