A car of mass m1 = 2000.0kg is moving at speed v1i = 20.0m/s towards East. A truck of mass m2 = 5000.0kg is moving at speed v2i = 10.0m/s towards North. They collide at an intersection and get entangled (complete inelastic collision).
1. What is the magnitude and direction of the final velocity of the entangled automobiles?
2. How much kinetic energy is lost in the collision. That is, calculate the change in the kinetic energy of the system
Me = East momentum = 2000*20
Mn = North momentum = 5000*10
that remains the same so after
Me = 7000 Veast = 2000*20 so Ve = 40/7
Mn = 7000 Vnorth = 5000 *10 so Vn = 50/7
|v| = (1/7)sqrt(40^2+50^2)
tan angle east of north = Ve/Vn
Ke before = (1/2)2000*400 + (1/2)5000*100
Ke after = (1/2) 7000 * |v|^2
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
1. The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
m1 * v1i + m2 * v2i = (m1 + m2) * vf
Substituting the given values:
2000.0kg * 20.0m/s + 5000.0kg * 10.0m/s = (2000.0kg + 5000.0kg) * vf
Now solve for vf:
40000.0kg*m/s + 50000.0kg*m/s = 7000.0kg * vf
90000.0kg*m/s = 7000.0kg * vf
vf = 90000.0kg*m/s / 7000.0kg
vf ≈ 12.8571m/s
The magnitude of the final velocity is approximately 12.8571 m/s. To determine the direction, we can use the principle of vector addition. Since the car and truck were initially moving in different directions, the final velocity will be a combination of their velocities.
To find the direction, we can use trigonometry. The car was initially moving towards the East and the truck was moving towards the North. This forms a right-angled triangle. Using the Pythagorean theorem, we can find the angle between the final velocity and the East direction:
tan(θ) = opposite / adjacent
tan(θ) = 20.0m/s / 10.0m/s
tan(θ) = 2
Taking the inverse tangent of both sides to find the angle:
θ = arctan(2)
θ ≈ 63.43 degrees
Therefore, the final velocity is approximately 12.8571 m/s, at an angle of approximately 63.43 degrees North of East.
2. To calculate the change in kinetic energy, we need to know the initial and final kinetic energies of the system. The initial kinetic energy can be calculated using the formula:
KE_initial = (1/2) * (m1 * v1i^2 + m2 * v2i^2)
Substituting the given values:
KE_initial = (1/2) * (2000.0kg * (20.0m/s)^2 + 5000.0kg * (10.0m/s)^2)
Now calculate KE_initial:
KE_initial ≈ 1,200,000 Joules
The final kinetic energy can be calculated using the formula:
KE_final = (1/2) * (m1 + m2) * vf^2
Substituting the known values:
KE_final = (1/2) * (2000.0kg + 5000.0kg) * (12.8571m/s)^2
Now calculate KE_final:
KE_final ≈ 410,628.57 Joules
Finally, the change in kinetic energy can be calculated as:
Change in KE = KE_initial - KE_final
Change in KE ≈ 1,200,000 Joules - 410,628.57 Joules
Change in KE ≈ 789,371.43 Joules
Therefore, the change in kinetic energy during the collision is approximately 789,371.43 Joules.