Car A with a mass of 1000kg is travelling east at 25m/s when it collides with car B, 2000kg, travelling north at 15m/s. After the collision, the two cars stick together and have the same velocity. Assume that they slide without friction after collision.
(a) What is the final velocity of the combined mass of cars (magnitude and direction)?
(b) How much kinetic energy was lost in this process and where did it go?
a. M1V1 + M2V2 = M1V + M2V.
1000*25 + 2000*15i = 1000V + 2000V,
25,000 + 30,000i = 3000V,
Divide by 1,000:
25 + 30i = 3V, V = 13m/s[50.2o] N. of E.
b. KE1 = 0.5M1*V1^2 + 0.5M2*V2^2,
KE1 = 500*25^2 + (1000*15i^2),
KE1 = 312,500 + 225,000 = 537,500 J. = KE before collision.
KE2 = 0.5M1*V + 0.5M2*V.
KE2 = 500*13^2 + 1000*13^2 = 253,500 J. = KE after collision.
KE1-KE2 = KE lost after collision.
To solve this problem, we can first calculate the momentum of each car before the collision. We can then use the conservation of momentum to find the final velocity of the combined mass of the two cars. Finally, we can calculate the change in kinetic energy to determine how much energy was lost in the collision.
(a) Final velocity of the combined mass:
Before the collision, we have the following information about the velocities:
Car A: mass (m1) = 1000 kg, velocity (v1) = 25 m/s to the east
Car B: mass (m2) = 2000 kg, velocity (v2) = 15 m/s to the north
To calculate the momentum before the collision:
Momentum of car A (p1) = mass of car A (m1) x velocity of car A (v1)
= 1000 kg x 25 m/s = 25000 kg·m/s to the east
Momentum of car B (p2) = mass of car B (m2) x velocity of car B (v2)
= 2000 kg x 15 m/s = 30000 kg·m/s to the north
Since momentum is a vector quantity, both p1 and p2 have different directions. To find the resultant momentum after the collision, we can add these vectors together.
To find the magnitude and direction of the final velocity of the combined mass, we can use the Pythagorean theorem and trigonometry.
Resultant momentum (p) = sqrt( p1^2 + p2^2 ) (magnitude)
= sqrt( (25000 kg·m/s)^2 + (30000 kg·m/s)^2 )
= sqrt( 625000000 kg^2·m^2/s^2 + 900000000 kg^2·m^2/s^2 )
= sqrt( 1525000000 kg^2·m^2/s^2 )
= 1234000 kg·m/s (approximately)
The direction of the resultant momentum can be found using trigonometry. Since the two cars stick together and move with the same velocity after the collision, the direction is the same as that of the initial momentum of car A.
Therefore, the final velocity of the combined mass is approximately 1234000 kg·m/s to the east.
(b) Kinetic energy lost and where it went:
To calculate the kinetic energy lost in the collision, we can compare the kinetic energy before and after the collision.
Before the collision, the total kinetic energy of the system is given by:
Kinetic energy before collision = 1/2 x mass of car A x (velocity of car A)^2 + 1/2 x mass of car B x (velocity of car B)^2
= 1/2 x 1000 kg x (25 m/s)^2 + 1/2 x 2000 kg x (15 m/s)^2
= 312500 J + 225000 J
= 537500 J (approximately)
After the collision, the two cars stick together and have the same velocity. So, the final kinetic energy is given by:
Kinetic energy after collision = 1/2 x mass of combined mass x (final velocity)^2
= 1/2 x (1000 kg + 2000 kg) x (1234 m/s)^2
= 1/2 x 3000 kg x (1234 m/s)^2
= 1/2 x 3000 kg x 1522756 m^2/s^2
= 2284134000 J (approximately)
Therefore, the kinetic energy lost in the process is:
Kinetic energy lost = Kinetic energy before collision - Kinetic energy after collision
= 537500 J - 2284134000 J
= -1746636500 J
The negative sign indicates that the kinetic energy was lost from the system. This energy is dissipated in the form of heat, sound, and deformation of the cars during the collision.
To solve this problem, we need to apply the principles of conservation of momentum and conservation of kinetic energy.
(a) Final velocity of the combined mass of cars (magnitude and direction):
Step 1: Calculate the initial momentum of each car.
The momentum of an object is given by the formula: momentum = mass × velocity.
For car A:
Momentum of car A = 1000 kg × 25 m/s = 25000 kg·m/s (east)
For car B:
Momentum of car B = 2000 kg × 15 m/s = 30000 kg·m/s (north)
Step 2: Use the principle of conservation of momentum.
Since there is no external force acting on the system, the total momentum before the collision is equal to the total momentum after the collision.
Total initial momentum = Total final momentum
In this case, the final momentum is the momentum of the combined mass of the two cars after sticking together.
Total initial momentum = Momentum of car A + Momentum of car B = 25000 kg·m/s (east) + 30000 kg·m/s (north)
To find the magnitude and direction of the final velocity, we can use the Pythagorean theorem. The direction can be determined using trigonometry.
Magnitude of the final velocity = √[(final momentum in x-direction)² + (final momentum in y-direction)²]
Direction of the final velocity = arctan(final momentum in y-direction / final momentum in x-direction)
Step 3: Calculate the magnitude and direction of the final velocity.
The magnitude of the final velocity is given by:
√[(25000 kg·m/s)² + (30000 kg·m/s)²] = √(625000000 + 900000000) kg·m²/s² = √1525000000 kg·m²/s² ≈ 39044.73 kg·m²/s²
The direction of the final velocity can be calculated as:
arctan(30000 kg·m/s / 25000 kg·m/s) = arctan(1.2) ≈ 50.19° north of east
Therefore, the final velocity of the combined mass of the cars is approximately 39044.73 kg·m²/s² at an angle of 50.19° north of east.
(b) How much kinetic energy was lost in this process and where did it go?
Step 1: Calculate the initial kinetic energy of each car.
The kinetic energy of an object is given by the formula: kinetic energy = 1/2 × mass × velocity².
For car A:
Kinetic energy of car A = 1/2 × 1000 kg × (25 m/s)² = 312500 J
For car B:
Kinetic energy of car B = 1/2 × 2000 kg × (15 m/s)² = 225000 J
Step 2: Use the principle of conservation of kinetic energy.
The total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Total initial kinetic energy = Total final kinetic energy
In this case, the final kinetic energy is the kinetic energy of the combined mass of the two cars after sticking together.
Total initial kinetic energy = Kinetic energy of car A + Kinetic energy of car B = 312500 J + 225000 J
Step 3: Calculate the lost kinetic energy.
Lost kinetic energy = Total initial kinetic energy - Total final kinetic energy
Lost kinetic energy = (312500 J + 225000 J) - Total final kinetic energy
The lost kinetic energy in this process went into other forms of energy, such as deformation of the cars, heat due to friction, and sound.
Note: The actual calculation for the lost kinetic energy and its distribution would require more information about the collision, such as the coefficients of restitution and the change in kinetic energy during the collision.