An 89.5 kg fullback moving east with a speed of 5.7 m/s is tackled by a 81.5 kg opponent running west at 2.58 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
LOL, they clung to each other after so after the crash the total mass is
89.5 + 81.5 = 171 kg
now east, the fullback I call +x
initial momentum
= 89.5*5.7 - 81.5*2.58
= 431 kg m/s
final momentum
= 171*v
Newton #1 say those are identical so
171 v = 431
v = 2.52 m/s
To calculate the velocity of the players just after the tackle, we can use the law of conservation of momentum. According to this law, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of a body is calculated by multiplying its mass by its velocity.
Let's assume the fullback moving east is player 1 and the opponent running west is player 2.
The initial momentum of player 1 (P1i) is calculated as follows:
P1i = mass of player 1 × velocity of player 1
= 89.5 kg × 5.7 m/s
= 510.15 kg·m/s
The initial momentum of player 2 (P2i) is calculated as follows:
P2i = mass of player 2 × velocity of player 2
= 81.5 kg × (-2.58 m/s) [Note: Velocity is negative because player 2 is moving in the opposite direction]
= -210.87 kg·m/s
Now, the total initial momentum (Pi) is the sum of P1i and P2i:
Pi = P1i + P2i
= 510.15 kg·m/s - 210.87 kg·m/s
= 299.28 kg·m/s
Since the collision is perfectly inelastic, the players stick together after the collision, moving with a common velocity.
Let's assume the common velocity of both players after the tackle is V.
The total momentum after the collision (Pf) is calculated as follows:
Pf = (mass of player 1 + mass of player 2) × V
= (89.5 kg + 81.5 kg) × V
= 171 kg × V
According to the law of conservation of momentum, Pi = Pf.
So, we can equate the initial and final momentum to find the value of V:
299.28 kg·m/s = 171 kg × V
Now, we can solve for V:
V = 299.28 kg·m/s / 171 kg
V = 1.75 m/s
Therefore, the velocity of the players just after the tackle is 1.75 m/s. This is the answer to part (a).
To calculate the decrease in kinetic energy during the collision, we need to find the initial and final kinetic energies and then subtract the final kinetic energy from the initial kinetic energy.
The initial kinetic energy (Ki) is calculated by using the formula:
Ki = (1/2) × mass of player 1 × (velocity of player 1)^2 + (1/2) × mass of player 2 × (velocity of player 2)^2
Ki = (1/2) × 89.5 kg × (5.7 m/s)^2 + (1/2) × 81.5 kg × (2.58 m/s)^2
Ki = 1291.83 J
The final kinetic energy (Kf) is calculated using the formula:
Kf = (1/2) × (mass of player 1 + mass of player 2) × (velocity of the players just after the tackle)^2
Kf = (1/2) × 171 kg × (1.75 m/s)^2
Kf = 202.54 J
Therefore, the decrease in kinetic energy during the collision is Ki - Kf:
ΔK = Ki - Kf = 1291.83 J - 202.54 J = 1089.29 J
Hence, the decrease in kinetic energy during the collision is 1089.29 J. This is the answer to part (b).
To solve this problem, we can use the conservation of momentum and the conservation of kinetic energy.
(a) First, let's calculate the velocity of the players just after the tackle.
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The momentum (p) is calculated by multiplying the mass (m) by the velocity (v):
Momentum (p) = mass (m) × velocity (v)
So, before the collision, the momentum of the fullback moving east (player 1) is:
Momentum1 (p1) = mass1 (m1) × velocity1 (v1)
= 89.5 kg × 5.7 m/s
And the momentum of the opponent running west (player 2) is:
Momentum2 (p2) = mass2 (m2) × velocity2 (v2)
= 81.5 kg × -2.58 m/s (since the opponent is moving west, we use a negative sign)
Since the collision is perfectly inelastic, the two players stick together after the tackle. Therefore, their final velocity will be the same.
The total mass after the collision (m_total) is the sum of the masses of both players:
m_total = m1 + m2
Using the conservation of momentum, we can set up an equation:
m_total × final velocity = p1 + p2
Substituting the known values, we get:
(m1 + m2) × final velocity = m1 × v1 + m2 × v2
Now we can calculate the final velocity:
(89.5 kg + 81.5 kg) × final velocity = (89.5 kg × 5.7 m/s) + (81.5 kg × -2.58 m/s)
Solving for the final velocity will give us the answer to part (a).
(b) Next, let's calculate the decrease in kinetic energy during the collision.
The initial kinetic energy of both players can be calculated using the equation:
Kinetic energy (KE) = (1/2) × mass × velocity^2
So, before the collision, the initial kinetic energy of the fullback moving east (player 1) is:
KE1 = (1/2) × mass1 × velocity1^2
And the initial kinetic energy of the opponent running west (player 2) is:
KE2 = (1/2) × mass2 × velocity2^2
The total initial kinetic energy before the collision (KE_initial_total) is the sum of the kinetic energies of both players:
KE_initial_total = KE1 + KE2
After the collision, the players stick together and the final kinetic energy (KE_final) is:
KE_final = (1/2) × m_total × final velocity^2
The decrease in kinetic energy during the collision (ΔKE) is:
ΔKE = KE_initial_total - KE_final
Solving for ΔKE will give us the answer to part (b).
initial ke = (1/2)89.5(5.7)^2 + (1/2)81.5(2.58)^2
final ke = (1/2) 171 (v)^2