13. A red car of mass m is heading north (direction 0°). It collides at an intersection with a yellow car of mass 1.3m heading east (direction 90°). Immediately after the collision, the cars lock together and travel at in direction 42°. What is the speed of the yellow car just before the collision?
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To find the speed of the yellow car just before the collision, we can use the principles of conservation of momentum and conservation of angular momentum.
1. Conservation of Momentum:
Before the collision, we have two cars, a red car and a yellow car, traveling in different directions. The total momentum (linear momentum) of the system before the collision is given by the vector sum of the momenta of the red and yellow cars.
Since the red car is heading north (direction 0°), its momentum can be written as m * v_red * [0°], where v_red is the speed of the red car.
The yellow car is heading east (direction 90°), so its momentum can be written as 1.3m * v_yellow * [90°], where v_yellow is the speed of the yellow car.
The vector sum of these momenta should combine to equal the total momentum of the system after the collision when the cars lock together.
2. Conservation of Angular Momentum:
When the cars lock together and travel in a new direction (42°), the total angular momentum of the system is conserved. Since the cars lock together, we can assume that the moment arm distance between the center of mass and the axis of rotation is zero. Therefore, the angular momentum of the system remains unchanged.
3. Finding the Speed of the Yellow Car:
To solve for the speed of the yellow car just before the collision, we set up equations based on the principles above.
Using conservation of momentum:
m * v_red * [0°] + 1.3m * v_yellow * [90°] = (m + 1.3m) * v_total * [42°]
Using conservation of angular momentum:
m * v_red * [0°] * 0 + 1.3m * v_yellow * [90°] * 0 = (m + 1.3m) * v_total * [42°] * d
Solving these equations will give you the speed of the yellow car just before the collision.