Two automobiles, each of mass 800kg , are moving at the same speed, 18m/s , when they collide and stick together. At what speed does the wreckage move if one car was driving north and one east.
initial momentum
North 800 * 18
East 800 * 18
final momentum
North 1600 v sin T
East 1600 v cos T
(I am pretending I do not see that V = 18 sqrt 2 and T = 45 degrees)
800 * 18 = 1600 v sin T
800 * 18 = 1600 v cos T
so sin T = cos T = 1/sqrt 2
That is T = 45 degrees
so
800 * 18 = 1600 v /sqrt2
800 * 18 = 800 v /sqrt 2
v = 18 sqrt 2
v = 18 sqrt 2
To determine the final velocity of the wreckage after the collision, we can use the principle of conservation of momentum.
Momentum is defined as the product of an object's mass and its velocity. Mathematically, it can be expressed as:
Momentum = mass * velocity
According to the conservation of momentum principle, the total momentum before the collision should be equal to the total momentum after the collision.
Let's break down the initial momentum in the x and y directions:
For the car moving north (vertical direction):
Momentum before = mass * velocity
Momentum in the y-direction before = 800 kg * 18 m/s = 14,400 kg·m/s
For the car moving east (horizontal direction):
Momentum before = mass * velocity
Momentum in the x-direction before = 800 kg * 18 m/s = 14,400 kg·m/s
Now, since the cars collide and stick together, the final mass of the wreckage will be the sum of the masses of the two cars:
Final mass = 800 kg + 800 kg = 1600 kg
To find the final velocity of the wreckage, let's calculate the total momentum after the collision:
Total momentum after = Final mass * Final velocity
Since the wreckage moves in an unknown direction, we can break down its velocity into components using vector addition:
Let's say the final velocity of the wreckage is v_wreckage m/s, and the angle it makes with the positive x-axis is θ.
Horizontal velocity component (x-direction): v_wreckage * cos(θ)
Vertical velocity component (y-direction): v_wreckage * sin(θ)
Now, using the conservation of momentum:
Total momentum after = Momentum in x-direction after + Momentum in y-direction after
Final mass * Final velocity = (Final mass) * (v_wreckage * cos(θ)) + (Final mass) * (v_wreckage * sin(θ))
1600 kg * v_wreckage = (1600 kg) * (v_wreckage * cos(θ)) + (1600 kg) * (v_wreckage * sin(θ))
Now, we can simplify the equation:
1600 kg * v_wreckage = (1600 kg * v_wreckage * cos(θ)) + (1600 kg * v_wreckage * sin(θ))
Dividing through by 1600 kg * v_wreckage:
1 = cos(θ) + sin(θ)
Since cos(θ) + sin(θ) = √2/2 for θ = 45 degrees, we can conclude that:
1 = √2/2 + √2/2
1 = √2
Simplifying further:
√2 = 1
This is not possible, so there is no solution for the case where the cars collide and stick together. This indicates that there might be an error or oversight in the given information or assumptions.
To find the speed at which the wreckage moves after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
Before the collision, each car has its momentum, which can be calculated as the product of its mass and velocity. Let's calculate the momentum of each car:
Car 1 momentum = mass × velocity = 800 kg × 18 m/s
Car 2 momentum = mass × velocity = 800 kg × 18 m/s
Since the two cars collide and stick together, the total mass of the wreckage will be the sum of the masses of both cars:
Total mass of wreckage = mass of Car 1 + mass of Car 2 = 800 kg + 800 kg
The total momentum after the collision will be equal to the momentum of the wreckage, which can be calculated using the total mass and the final velocity of the wreckage. Let's call the final velocity of the wreckage V.
Total momentum after the collision = Total mass of wreckage × V
Since momentum is conserved, we can equate the total momentum before the collision to the total momentum after the collision:
Car 1 momentum + Car 2 momentum = Total momentum after the collision
800 kg × 18 m/s + 800 kg × 18 m/s = (800 kg + 800 kg) × V
From this equation, we can solve for V, which will give us the speed at which the wreckage moves.