A 79.0-kg fullback running east with a speed of 4.00 m/s is tackled by a 93.0-kg opponent running north with a speed of 3.00 m/s.
(a) Why does the tackle constitute a perfectly inelastic collision?
(b) Calculate the velocity of the players immediately after the tackle.
magnitude m/s
direction ° north of east
(c) Determine the mechanical energy that is lost as a result of the collision.
J
(d) Where did the lost energy go?
(a) The tackle constitutes a perfectly inelastic collision because the two players stick together and move as a single object after the collision. In an inelastic collision, kinetic energy is not conserved, and the objects may deform or stick together.
To calculate the velocity of the players immediately after the tackle, we can use the law of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
(b) To find the velocity after the tackle, we need to find the components of the initial velocities for both players.
The fullback's velocity (v1) can be broken down into its eastward (x-direction) and northward (y-direction) components:
v1x = 4.00 m/s (eastward)
v1y = 0 m/s (no northward component)
The opponent's velocity (v2) can also be broken down into components:
v2x = 0 m/s (no eastward component)
v2y = 3.00 m/s (northward)
After the tackle, the players move as a single object, so we can find their combined velocity (vf) using conservation of momentum:
(v1 + v2)f = vf
Using the law of conservation of momentum:
(m1 * v1x + m2 * v2x) / (m1 + m2) = vf in the x-direction
(m1 * v1y + m2 * v2y) / (m1 + m2) = vf in the y-direction
Where m1 is the mass of the fullback (79.0 kg) and m2 is the mass of the opponent (93.0 kg).
Substituting in the given values, we get:
(79.0 kg * 4.00 m/s + 93.0 kg * 0 m/s) / (79.0 kg + 93.0 kg) = vf in the x-direction
Simplifying the equation:
316.0 kg*m/s / 172.0 kg = vf in the x-direction
The resulting velocity in the x-direction (east) is:
vf in the x-direction = 1.8372 m/s (rounded to 4 decimal places)
To find the direction of the velocity, we can calculate the angle (θ) north of east:
θ = tan^(-1)(vfy / vfx)
Substituting the values, we get:
θ = tan^(-1)((79.0 kg * 0 m/s + 93.0 kg * 3.00 m/s) / (79.0 kg * 4.00 m/s + 93.0 kg * 0 m/s))
Simplifying the equation:
θ = tan^(-1)(279.0 kg*m/s / 316.0 kg*m/s)
The resulting angle is:
θ = 40.49° (rounded to 2 decimal places)
Therefore, the velocity of the players immediately after the tackle is 1.8372 m/s at an angle of 40.49° north of east.
(c) To determine the mechanical energy lost as a result of the collision, we need to find the initial and final kinetic energies.
The initial kinetic energy is the sum of the kinetic energies of the fullback and the opponent before the collision:
KE_initial = (1/2) * m1 * (v1^2) + (1/2) * m2 * (v2^2)
Substituting the given values, we get:
KE_initial = (1/2) * 79.0 kg * (4.00 m/s)^2 + (1/2) * 93.0 kg * (3.00 m/s)^2
Simplifying the equation:
KE_initial = 316.0 J + 1309.5 J
KE_initial = 1625.5 J
The final kinetic energy is the kinetic energy of the players moving as a single object after the tackle:
KE_final = (1/2) * (m1 + m2) * (vf^2)
Substituting the calculated value for vf and the given values for m1 and m2, we get:
KE_final = (1/2) * (79.0 kg + 93.0 kg) * (1.8372 m/s)^2
Simplifying the equation:
KE_final = (1/2) * 172.0 kg * (3.3769 m/s)^2
KE_final = 290.4 J
The mechanical energy lost as a result of the collision is:
Energy_lost = KE_initial - KE_final
Energy_lost = 1625.5 J - 290.4 J
Energy_lost = 1335.1 J
Therefore, the mechanical energy lost as a result of the collision is 1335.1 Joules.
(d) The lost energy is converted into other forms, such as sound, heat, or deformation of the players' bodies. During the tackle, the kinetic energy is transformed into these other forms, resulting in the loss of mechanical energy.