A 1365-kg car moving north at 27.0 m/s is struck by a 2165-kg car moving east at 12.0 m/s. The cars are stuck together. How fast and in what direction do they move immediately after the collision?
To find the velocity and direction of the two cars immediately after the collision, we can use the principles of conservation of momentum.
Step 1: Find the momentum of each car before the collision.
The momentum (p) of an object is given by its mass (m) multiplied by its velocity (v). Using this formula:
Momentum of the first car (north) = mass of the first car × velocity of the first car
= 1365 kg × 27.0 m/s
= 36855 kg·m/s
Momentum of the second car (east) = mass of the second car × velocity of the second car
= 2165 kg × 12.0 m/s
= 25980 kg·m/s
Step 2: Find the total momentum before the collision.
The total momentum before the collision is the sum of the momenta of both cars.
Total momentum before the collision = Momentum of the first car + Momentum of the second car
= 36855 kg·m/s + 25980 kg·m/s
= 62835 kg·m/s
Step 3: Find the velocity and direction after the collision.
Since the cars stick together after the collision, we can consider them as one system. The total mass after the collision is the sum of the masses of both cars.
Total mass after the collision = mass of the first car + mass of the second car
= 1365 kg + 2165 kg
= 3530 kg
To find the velocity after the collision, we divide the total momentum before the collision by the total mass after the collision.
Velocity after the collision = Total momentum before the collision / Total mass after the collision
= 62835 kg·m/s / 3530 kg
= 17.80 m/s
To find the direction, we use the Pythagorean theorem and trigonometry. The direction can be defined by the angle θ between the direction of the first car (north) and the resultant velocity after the collision.
θ = arctan(Vy / Vx)
where Vy is the vertical component of the velocity and Vx is the horizontal component of the velocity.
Since the first car is moving north, the vertical component of the velocity is equal to the velocity after the collision, and the horizontal component is equal to 0.
θ = arctan(17.80 / 0) = undefined
The resultant velocity after the collision is approximately 17.80 m/s in an undefined direction, since we cannot determine an angle when dividing by 0.
To determine how fast and in what direction the cars move immediately after the collision, we can use the principles of conservation of momentum.
Conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision (assuming no external forces are acting on the system).
Let's label the car moving north as car A and the car moving east as car B.
The momentum of an object is calculated by multiplying its mass with its velocity. Therefore, the momentum of car A before the collision is given by:
Momentum of car A (before collision) = mass of car A x velocity of car A
= 1365 kg x 27.0 m/s (north)
= 36855 kg·m/s (north)
Similarly, the momentum of car B before the collision is:
Momentum of car B (before collision) = mass of car B x velocity of car B
= 2165 kg x 12.0 m/s (east)
= 25980 kg·m/s (east)
Since momentum is a vector quantity, we need to consider both the magnitude and direction when we add them up.
The total momentum before the collision is the vector sum of the individual momenta:
Total momentum before collision = Momentum of car A (before collision) + Momentum of car B (before collision)
= 36855 kg·m/s (north) + 25980 kg·m/s (east)
To find the net momentum after the collision, we need to find the resultant momentum of the two cars stuck together. Since the cars are stuck together, they move as one unit.
Let's label the resultant momentum as "Total momentum after collision."
Using the principle of conservation of momentum:
Total momentum before collision = Total momentum after collision
So,
36855 kg·m/s (north) + 25980 kg·m/s (east) = Total momentum after collision
To find the magnitude and direction of the total momentum after the collision, we can use the Pythagorean theorem and trigonometry.
The magnitude of the total momentum is given by:
Magnitude of total momentum after collision = √[(Total momentum after collision)^2]
The direction of the total momentum is given by the angle formed with the positive x-axis.
To find the angle, we can use the inverse tangent function:
Angle = arctan[(Total momentum after collision in the y-direction) / (Total momentum after collision in the x-direction)]
Given that the cars are stuck together after the collision, their masses combine, resulting in a single mass for the system.
Total mass = mass of car A + mass of car B
= 1365 kg + 2165 kg
= 3530 kg
Therefore,
Total momentum after collision = Total mass x final velocity
= 3530 kg x final velocity
Now, we can calculate the magnitude and direction of the total momentum after the collision.