A 450 kg car traveling north at 30 m/s collides with a 560 kg car traveling west at 35 m/s. If after the collision they become entangled and move off together, (a) what is magnitude of the final momentum of the cars? (b) What is their final speed of the cars? (c) What is their direction relative to the -x axis?
I understand that this specific word problem is an Inelastic equation, and I already have made my chart for object A, and object B. I'm just not sure what to do next.
M1*V1 + M2*V2 = M1*V + M2*V
450*30i + 560*(-35) = 450V + 560V
13,500i - 19,600 = 1,010V
b. V = -19.41 + 13.37i
V = sqrt = (19.41^2+13.37^2) = 23.6 m/s.
a. Momentum = M1*V + M2*V =
450*23.6 + 560*23.6 = 23,836 kg-m/s.
c. Tan A = Y/X = 13.71/-19.41=-0.70634
A = -35.2o = 35.2o CW from -x axis =
35.2o N. of W.
To solve this problem, we need to apply the principles of conservation of momentum and analyze the vector components involved.
Step 1: Set up your coordinate system.
Let the east-west direction be the x-axis and the north-south direction be the y-axis. The positive x-axis points east, and the positive y-axis points north. We'll denote the final velocity of the cars as vΣ, x-component as vΣx, and y-component as vΣy.
Step 2: Calculate the initial momentum of the cars.
Momentum (p) is given by the product of mass (m) and velocity (v): p = m * v.
For the 450 kg car traveling north:
m1 = 450 kg (mass)
v1 = +30 m/s (velocity, as it is traveling north)
p1 = m1 * v1
For the 560 kg car traveling west:
m2 = 560 kg (mass)
v2 = -35 m/s (velocity, as it is traveling west)
p2 = m2 * v2
Step 3: Calculate the momentum components.
Since momentum is a vector quantity, it has both magnitude and direction. We need to break down the initial momentum vectors into their x and y components.
For the 450 kg car:
p1x = 0 (no x-component as it is traveling north)
p1y = p1
For the 560 kg car:
p2x = p2 (since the car is traveling west, the whole momentum is in the x-axis)
p2y = 0 (no y-component as it is traveling west)
Step 4: Calculate the final momentum.
Since the cars become entangled and move off together, their combined mass is mΣ = m1 + m2, and their final velocity is vΣ.
Using the conservation of momentum principle:
mΣ * vΣ = p1 + p2
Step 5: Calculate the final velocity components.
We can now find the x and y components of the final velocity using the formula:
vΣx = (p1x + p2x) / mΣ
vΣy = (p1y + p2y) / mΣ
Step 6: Calculate the magnitude and direction of the final momentum and velocity.
To find the magnitude of the final momentum, use the Pythagorean theorem:
|vΣ| = sqrt(vΣx^2 + vΣy^2)
To find the direction relative to the -x-axis, use trigonometry:
θ = tan^(-1)(vΣy / vΣx)
So, to answer the specific points of the question:
(a) The magnitude of the final momentum of the cars is |vΣ|.
(b) The final speed of the cars is |vΣ|.
(c) The direction relative to the -x-axis is given by θ.
Plug in the values for each step, and you should be able to find the solution to this problem.