Two cars collide at an intersection. Car A, with a mass of 2.0x103 kg, is going from west to east, while car B, of mass 1500 kg, is going from north to south at 15 m/s. As a result of this collision, the two cars become enmeshed and move as one afterward. In your role as an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle 65 south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car A going just before the collision?
(a) Well, it seems like those cars really decided to tango, huh? To find the speed of the enmeshed cars just after the collision, we can use the principle of conservation of momentum. The total momentum of the system before the collision should be equal to the total momentum after the collision.
Now, car A's momentum before the collision is given by its mass (2.0x10^3 kg) multiplied by its velocity (let's call it v). Car B's momentum, on the other hand, is given by its mass (1.5x10^3 kg) multiplied by its velocity (15 m/s). Since the cars become enmeshed and move as one afterward, their combined momentum just after the collision should be equal to the total momentum before the collision.
So, we can write the equation as:
(2.0x10^3 kg) * v + (1.5x10^3 kg) * (15 m/s) = total momentum after the collision.
Now, to find the magnitude of the enmeshed cars' velocity, you just need to solve that equation. Remember, though, we also know the angle of their velocity from the point of impact, so it will be a bit trickier to find the actual speed.
(b) To determine the speed of car A just before the collision, we unfortunately don't have enough information. The velocity of car A just before the collision is unknown and not given in the problem. It seems like car A is keeping it a secret!
I hope I was able to shed some light on the situation. Remember, when cars collide, comedy does not necessarily ensue. Stay safe on the roads!
To solve this problem, we can use the conservation of momentum principle. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
(a) To calculate the speed of the enmeshed cars just after the collision, we need to find their combined mass and momentum. Let's denote the final speed of the enmeshed cars as v.
The momentum before the collision is given by:
Momentum of Car A before collision = mass of Car A × velocity of Car A
= (2.0 x 10^3 kg) × velocity of Car A
Momentum of Car B before collision = mass of Car B × velocity of Car B
= (1500 kg) × (15 m/s)
The momentum after the collision is given by:
Momentum of enmeshed cars after collision = (mass of Car A + mass of Car B) × velocity of enmeshed cars
= (2.0 x 10^3 kg + 1500 kg) × v
According to the conservation of momentum principle, the total momentum before the collision is equal to the total momentum after the collision:
Momentum of Car A before collision + Momentum of Car B before collision = Momentum of enmeshed cars after collision
Therefore,
(2.0 x 10^3 kg) × velocity of Car A + (1500 kg) × (15 m/s) = (2.0 x 10^3 kg + 1500 kg) × v
Now we can solve for v by rearranging the equation:
v = [(2.0 x 10^3 kg) × velocity of Car A + (1500 kg) × (15 m/s)] / [(2.0 x 10^3 kg + 1500 kg)]
(b) To calculate the speed of Car A just before the collision, we can use the same equation from part (a) but solve for the velocity of Car A:
velocity of Car A = [(2.0 x 10^3 kg + 1500 kg) × v - (1500 kg) × (15 m/s)] / (2.0 x 10^3 kg)
Now we can substitute the given angle of 65 degrees south of east to find the velocity components of the enmeshed cars:
Velocity of enmeshed cars in the x-direction (east): vcos(65)
Velocity of enmeshed cars in the y-direction (south): -vsin(65)
Therefore, the speed of the enmeshed cars (v) just after the collision and the speed of Car A just before the collision (velocity of Car A) can be calculated by substituting values into the equations above.
To solve this problem, we'll need to use the principles of conservation of momentum and vector addition. Let's analyze the problem step by step:
(a) How fast were the enmeshed cars moving just after the collision?
1. We start by finding the initial momentum of car B in the north-south direction (y-axis):
Momentum of car B (initial) = mass of car B x velocity of car B
= 1500 kg x (15 m/s)
= 22,500 kg·m/s (north)
2. Next, we find the initial momentum of car A in the east-west direction (x-axis):
Momentum of car A (initial) = mass of car A x velocity of car A
= 2.0x10^3 kg x velocity of car A (unknown)
3. Since the two cars become enmeshed and move as one, the total momentum after the collision must be equal to the initial momentum before the collision. However, we need to consider the direction as well.
4. To find the total momentum just after the collision, we can break it down into its components:
Momentum of enmeshed cars (after) = momentum in the x-direction + momentum in the y-direction
5. We know the angle at which the enmeshed cars move after the collision is 65° south of east. Let's break this vector into its east-west (x-axis) and north-south (y-axis) components.
6. The x-component of the momentum (P_x) is given by the total momentum multiplied by the cosine of the angle:
P_x = momentum after collision x cos(angle)
= (momentum of car A (initial) + momentum of car B (initial)) x cos(65°)
7. The y-component of the momentum (P_y) is given by the total momentum multiplied by the sine of the angle:
P_y = momentum after collision x sin(angle)
= (momentum of car B (initial)) x sin(65°)
8. Now, let's substitute the values we have:
P_x = (2.0x10^3 kg x velocity of car A) x cos(65°)
P_y = (22,500 kg·m/s) x sin(65°)
9. Since the total momentum after the collision is the sum of the x and y components, we can find the magnitude (speed) by using the Pythagorean theorem:
Speed (enmeshed cars after) = sqrt(P_x^2 + P_y^2)
(b) How fast was car A going just before the collision?
To find the speed of car A just before the collision, we can use the same approach as above but with only car A's initial momentum (as car B comes to a stop):
Speed (car A before) = mass of car A x velocity of car A
By following these steps and performing the necessary calculations, we'll be able to determine the answers.