In a head-on collision of two cars with equal mass, if one car has no speed change while the other experiences a decrease in speed, we can analyze the total momentum before and after the collision.
Let's denote the mass of each car as "m" and their initial velocities as "v1" and "v2" (positive values indicating motion in one direction, negative values indicating motion in the opposite direction).
Before the collision: The total momentum is the sum of the individual momenta of the two cars. Since one car has no speed change, its momentum remains unchanged:
Momentum of car 1 (with no speed change) = m * v1
The other car experiences a decrease in speed, so its momentum changes:
Momentum of car 2 (with speed decrease) = m * (v2_f - v2_i)
Here, "v2_f" represents the final velocity of car 2 after the collision, and "v2_i" represents its initial velocity.
Total momentum before the collision = Momentum of car 1 + Momentum of car 2
After the collision: Assuming there are no external forces, momentum is conserved in a collision. Therefore, the total momentum after the collision should also be equal to the total momentum before the collision.
Total momentum after the collision = Total momentum before the collision
This means that the sum of the individual momenta after the collision should equal the sum of the individual momenta before the collision.
If we assume that car 1 has no speed change after the collision, its momentum remains the same:
Momentum of car 1 (after collision) = m * v1
For car 2, its velocity changes from "v2_i" to "v2_f" due to the decrease in speed. Therefore, its momentum after the collision is:
Momentum of car 2 (after collision) = m * (v2_f - v2_i)
Total momentum after the collision = Momentum of car 1 (after collision) + Momentum of car 2 (after collision)
Since the total momentum before and after the collision should be the same, we can equate the expressions:
Momentum of car 1 + Momentum of car 2 = Momentum of car 1 (after collision) + Momentum of car 2 (after collision)
m * v1 + m * (v2_f - v2_i) = m * v1 + m * (v2_f - v2_i)
As you can see, the mass "m" cancels out, and we are left with:
v1 + (v2_f - v2_i) = v1 + (v2_f - v2_i)
Therefore, the total momentum before and after the collision remains the same regardless of the speed changes.