A 1000 kg car moving at 27 m/s from left to right collides head-on with a 1500 kg car moving at 30 m/s right to left.
A) Determine the magnitude and direction of the velocity of the wreckage after the collision.
B) If a 1000 kg car moving at 36 m/s from left to right collides with a 1500 kg car moving at 30 m/s at 36.87 degrees N of W, what is the speed and direction of the wreckage?
To determine the magnitude and direction of the velocity of the wreckage after the collision in both scenario A and B, you can use the principles of conservation of linear momentum and apply them to the collision.
Scenario A:
We have two cars moving towards each other, so the direction of one car will be considered negative. Let's take the left-to-right direction as positive (+) and the right-to-left direction as negative (-).
1. Determine the initial momentum of each car:
Car 1 momentum = mass of car 1 * velocity of car 1
= 1000 kg * 27 m/s (left to right)
= 27000 kg m/s (+)
Car 2 momentum = mass of car 2 * velocity of car 2
= 1500 kg * (-30 m/s) (right to left)
= -45000 kg m/s (-)
2. Calculate the total initial momentum:
Total initial momentum = momentum of car 1 + momentum of car 2
= 27000 kg m/s + (-45000 kg m/s)
= -18000 kg m/s
3. Apply conservation of linear momentum:
The total momentum before the collision is equal to the total momentum after the collision.
m₁v₁ + m₂v₂ = m₃v₃ (where m represents mass and v represents velocity)
Since the two cars stick together after the collision, their combined mass is 1000 kg + 1500 kg = 2500 kg.
1000 kg * velocity of wreckage + 1500 kg * velocity of wreckage = 2500 kg * resulting velocity
Simplifying the equation:
(1000 kg + 1500 kg) * velocity of wreckage = 2500 kg * resulting velocity
2500 kg * velocity of wreckage = 2500 kg * resulting velocity
So, the magnitude of the velocity of the wreckage is the same as the resulting velocity.
4. Calculate the magnitude and direction of the velocity of the wreckage:
Given that the total initial momentum was -18000 kg m/s and that the resulting velocity will have the same magnitude,
magnitude of the resulting velocity = absolute value of (total initial momentum / resulting mass)
= absolute value of (-18000 kg m/s / 2500 kg)
= 7.2 m/s
The wreckage will have a magnitude of 7.2 m/s. Since the two cars were moving in opposite directions, the direction of the resulting velocity can be considered the direction of the car that was moving faster, which was the right-to-left direction in this case. So, the direction of the wreckage is to the left (-).
Scenario B:
To calculate the speed and direction of the wreckage in this scenario, we need to consider the angle of collision.
1. Determine the initial momentum of each car:
Car 1 momentum = mass of car 1 * velocity of car 1
= 1000 kg * 36 m/s (left to right)
= 36000 kg m/s (+)
Car 2 momentum = mass of car 2 * velocity of car 2
= 1500 kg * 30 m/s (36.87 degrees N of W)
= 45000 kg m/s * cos(36.87°) (towards West)
Note: We need to calculate the component of the momentum in the West direction using the angle.
2. Calculate the total initial momentum:
Total initial momentum = momentum of car 1 + momentum of car 2
= 36000 kg m/s + 45000 kg m/s * cos(36.87°) in the West direction
3. Apply conservation of linear momentum:
The total momentum before the collision is equal to the total momentum after the collision.
m₁v₁ + m₂v₂ = m₃v₃
Since the two cars stick together after the collision, their combined mass is 1000 kg + 1500 kg = 2500 kg.
1000 kg * velocity of wreckage + (1500 kg * velocity of wreckage * cosθ) = 2500 kg * resulting velocity
Simplifying the equation:
(1000 kg + 1500 kg * cosθ) * velocity of wreckage = 2500 kg * resulting velocity
So, the magnitude of the velocity of the wreckage is the same as the resulting velocity.
4. Calculate the magnitude and direction of the velocity of the wreckage:
Given that the total initial momentum is known, we can proceed to obtain the resulting velocity.
magnitude of the resulting velocity = absolute value of (total initial momentum / resulting mass)
= absolute value of (Total initial momentum / 2500 kg)
Note: The direction of the wreckage will depend on the angle θ. If the angle is 0° (East), the direction will be positive (+), and if it is 180° (West), the direction will be negative (-).
Plug in the given angle θ into the equation to obtain the speed and direction of the wreckage.