Two cars collide at an intersection. Car A was traveling east at 20 m/s, and Car B was traveling north at 15 m/s. After the collision, both cars stick together and move in a direction at 45 degrees to the east. Calculate their final velocity.
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To solve this problem, we can use the principle of conservation of momentum.
Step 1: Break down the initial velocities into their components.
Car A:
- Initial velocity in the x-direction (east) = 20 m/s
- Initial velocity in the y-direction (north) = 0 m/s
Car B:
- Initial velocity in the x-direction (east) = 0 m/s
- Initial velocity in the y-direction (north) = 15 m/s
Step 2: Calculate the total momentum before the collision.
Total momentum before collision = (mass of car A * velocity of car A) + (mass of car B * velocity of car B)
Since the masses of the cars are not given, assume they are equal, so the equation becomes:
Total momentum before collision = (mass * velocity of car A) + (mass * velocity of car B)
= mass * (velocity of car A + velocity of car B)
Total momentum before collision = mass * (20 m/s + 15 m/s)
= mass * 35 m/s
Step 3: Calculate the final velocity after the collision.
Assuming the cars stick together and move at 45 degrees to the east, the final velocity can be calculated by finding the x and y components of the final velocity and then combining them.
The magnitude of the final velocity, Vf can be calculated using the Pythagorean theorem:
Vf^2 = Vfx^2 + Vfy^2
Using trigonometry, we can determine that the final velocity components, Vfx and Vfy, are equal:
Vfx = Vfy = Vf * cos(45)
Hence, Vf * cos(45)^2 + Vf * cos(45)^2 = Vf^2
2 * Vf^2 * cos(45)^2 = Vf^2
2 * cos(45)^2 = 1
2 * (1/√2)^2 = 1
2 * (1/2) = 1
1 = 1
This shows that the magnitude of the final velocity, Vf is the same as the magnitude of the components, Vfx and Vfy.
Step 4: Calculate the magnitude of the final velocity.
Using conservation of momentum, the total momentum after the collision is equal to the total momentum before the collision:
Total momentum after collision = mass * Vfx
Total momentum after collision = mass * Vf * cos(45)
Total momentum after collision = mass * Vf * (1/√2)
Step 5: Set the total momentum before the collision equal to the total momentum after the collision:
mass * 35 m/s = mass * Vf * (1/√2)
35 m/s = Vf * (1/√2)
Solving for Vf, we get:
Vf = 35 m/s * √2
Vf ≈ 49.5 m/s
Therefore, the final velocity of the cars after the collision is approximately 49.5 m/s at an angle of 45 degrees to the east.
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