An 88.5 kg fullback moving east with a speed of 5.5 m/s is tackled by a 94.0 kg opponent running west at 2.90 m/s, and the collision is perfectly inelastic.
(a) Calculate the velocity of the players just after the tackle.
m/s
(b) Calculate the decrease in kinetic energy during the collision.
J
m₁v₁-m₂v₂=(m₁+m₂)u
u =(m₁v₁-m₂v₂)/(m₁+m₂)= …
ΔKE=m₁v₁²/2 + m₂v₂²/2 -(m₁+m₂)u²/2=..
To solve this problem, we can apply the principle of conservation of momentum and conservation of kinetic energy.
(a) To find the velocity of the players just after the tackle, we need to calculate the momentum of the system before and after the collision.
The momentum of an object is calculated by multiplying its mass by its velocity, using the formula:
momentum = mass x velocity
Before the collision, the momentum of the fullback (moving east) is given by:
momentum fullback = mass fullback x velocity fullback
= 88.5 kg x 5.5 m/s
= 486.75 kg·m/s (eastward)
The momentum of the opponent (running west) is given by:
momentum opponent = mass opponent x velocity opponent
= 94.0 kg x (-2.9 m/s) [Note: the velocity is negative because it's in the opposite direction]
= -273.1 kg·m/s (westward)
Since the collision is perfectly inelastic, the players stick together after the tackle, implying that their velocities are the same after the collision. Let's call this velocity "vf."
The total momentum after the collision is the sum of their individual momenta:
momentum total = momentum fullback + momentum opponent
= 486.75 kg·m/s (eastward) + (-273.1 kg·m/s) (westward)
= 213.65 kg·m/s (eastward)
Now, we can find their common velocity "vf" by dividing the total momentum by the combined mass of the two players:
momentum total = (mass fullback + mass opponent) x vf
Solving for vf, we get:
vf = momentum total / (mass fullback + mass opponent)
= 213.65 kg·m/s / (88.5 kg + 94.0 kg)
= 213.65 kg·m/s / 182.5 kg
≈ 1.17 m/s (eastward)
Therefore, the velocity of the players just after the tackle is approximately 1.17 m/s to the east.
(b) To calculate the decrease in kinetic energy during the collision, we can compare the initial and final kinetic energies.
Kinetic energy is given by the formula:
kinetic energy = 0.5 x mass x velocity^2
Before the collision, the fullback's initial kinetic energy is:
KEfullback_initial = 0.5 x mass fullback x (velocity fullback)^2
= 0.5 x 88.5 kg x (5.5 m/s)^2
= 1356.75 J
The opponent's initial kinetic energy is:
KEopponent_initial = 0.5 x mass opponent x (velocity opponent)^2
= 0.5 x 94.0 kg x (-2.9 m/s)^2
= 763.37 J
The total initial kinetic energy is the sum of their individual kinetic energies:
KE_total_initial = KEfullback_initial + KEopponent_initial
= 1356.75 J + 763.37 J
= 2120.12 J
After the collision, since the players stick together, they have a common velocity "vf" calculated earlier.
The final kinetic energy is thus:
KE_total_final = 0.5 x (mass fullback + mass opponent) x vf^2
= 0.5 x (88.5 kg + 94.0 kg) x (1.17 m/s)^2
≈ 174.53 J
The decrease in kinetic energy is the difference between the initial and final kinetic energies:
decrease in kinetic energy = KE_total_initial - KE_total_final
= 2120.12 J - 174.53 J
≈ 1945.59 J
Therefore, the decrease in kinetic energy during the collision is approximately 1945.59 J.