You are correct! This problem can be solved using the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision. In mathematical terms, we can write:
(m1 * v1_initial) + (m2 * v2_initial) = (m1 * v1_final) + (m2 * v2_final),
where:
- m1 and m2 are the masses of car #1 and car #2, respectively,
- v1_initial and v2_initial are the initial velocities of car #1 and car #2,
- v1_final and v2_final are the final velocities of car #1 and car #2.
Now let's plug in the given values and solve for the final velocities.
m1 = m2 = 254 kg
v1_initial = -2.5 m/s (since car #1 is moving west, its velocity is negative)
v2_initial = 3.7 m/s (since car #2 is moving east, its velocity is positive)
So the equation becomes:
(254 kg * -2.5 m/s) + (254 kg * 3.7 m/s) = (254 kg * v1_final) + (254 kg * v2_final).
Simplifying the equation, we get:
-635 kg m/s + 939.8 kg m/s = 254 kg * v1_final + 254 kg * v2_final.
303.8 kg m/s = 254 kg * v1_final + 254 kg * v2_final.
To solve for the final velocities, we can divide both sides of the equation by 254 kg:
(303.8 kg m/s) / (254 kg) = v1_final + v2_final.
1.196 m/s = v1_final + v2_final.
Now, we need to determine the individual final velocities. Since this is an elastic collision, the relative velocities before and after the collision will be the opposite.
v1_final = -v1_initial
v2_final = -v2_initial
Plugging in the values, we get:
v1_final = -(-2.5 m/s) = 2.5 m/s (since the negative signs cancel out)
v2_final = -(3.7 m/s) = -3.7 m/s.
Therefore, the final velocities after the collision are:
- Car #1 (moving west) has a final velocity of 2.5 m/s.
- Car #2 (moving east) has a final velocity of -3.7 m/s.