Mary is driving a car of mass 900 kilograms toward the north with a velocity of 22 meters/second. Danny is driving his 900-kilogram car with a velocity of 15 meters/second toward the east. They collide and both cars move together afterwards. What is the magnitude of the resultant momentum of the cars after the collision?
To find the magnitude of the resultant momentum of the cars after the collision, we need to calculate the total momentum before and after the collision.
Momentum is defined as the product of an object's mass and velocity. The formula for momentum is:
Momentum = mass * velocity
Given:
Mass of Mary's car (m1) = 900 kg
Velocity of Mary's car (v1) = 22 m/s
Mass of Danny's car (m2) = 900 kg
Velocity of Danny's car (v2) = 15 m/s
Before the collision, we need to calculate the total momentum of both cars separately using the momentum formula:
Momentum of Mary's car before the collision (p1) = m1 * v1
Momentum of Danny's car before the collision (p2) = m2 * v2
Now, let's calculate the momentum before collision:
p1 = 900 kg * 22 m/s = 19800 kg⋅m/s
p2 = 900 kg * 15 m/s = 13500 kg⋅m/s
The total momentum before the collision is the vector sum of the individual momenta:
Total momentum before collision = p1 + p2
Total momentum before collision = 19800 kg⋅m/s + 13500 kg⋅m/s = 33300 kg⋅m/s
After the collision, both cars move together, so the final velocity will be the same for both cars. Let's denote the final velocity as vf.
The total momentum after the collision is the sum of the individual momenta:
Total momentum after collision = (m1 + m2) * vf
Since both cars move together, the total mass after the collision is the sum of the individual masses:
Total mass after collision = m1 + m2 = 900 kg + 900 kg = 1800 kg
To find the final velocity (vf), we can use the principle of conservation of momentum, which states that the momentum before the collision is equal to the momentum after the collision. So,
Total momentum before collision = Total momentum after collision
33300 kg⋅m/s = 1800 kg * vf
Now, let's solve for vf:
vf = 33300 kg⋅m/s / 1800 kg ≈ 18.5 m/s
Now that we have the final velocity (vf), we can calculate the total momentum after the collision:
Total momentum after collision = (m1 + m2) * vf
Total momentum after collision = 1800 kg * 18.5 m/s = 33300 kg⋅m/s
Therefore, the magnitude of the resultant momentum of the cars after the collision is 33300 kg⋅m/s.