Two bumper cars crash head on in an elastic collision. One car #1 was moving 2.5 m/s west and car #2 was moving 3.7 m/s east. Each car has a mass of 254 kg. What are the final velocities after the collision?
I think this problem is supposed to be solved with m1v1, i + m2v2, i = m1v1, f + m2v2, f
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.
Yes, you are correct. The problem can be solved using the principle of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision. In equation form, this is expressed as:
m1v1i + m2v2i = m1v1f + m2v2f
where m1 and m2 are the masses of the two cars, v1i and v2i are the initial velocities, and v1f and v2f are the final velocities.
Given:
m1 = m2 = 254 kg (mass of both cars)
v1i = -2.5 m/s (negative sign indicates westward direction)
v2i = 3.7 m/s (positive sign indicates eastward direction)
To find the final velocities, we can solve the equation using algebra.
m1v1i + m2v2i = m1v1f + m2v2f
Substituting the given values:
(254 kg)(-2.5 m/s) + (254 kg)(3.7 m/s) = (254 kg)(v1f) + (254 kg)(v2f)
Simplifying the equation:
-635 kg·m/s + 938.8 kg·m/s = 254 kg(v1f + v2f)
303.8 kg·m/s = 254 kg (v1f + v2f)
Dividing both sides of the equation by 254 kg:
1.196 m/s = v1f + v2f
Now, we need to take into account the direction of the final velocities. Since the cars collide head-on, they will change direction after the collision. Car #1 was moving westward, so its final velocity will be negative. Car #2 was moving eastward, so its final velocity will be positive. Let's denote v1f as the final velocity of car #1 and v2f as the final velocity of car #2.
v1f + v2f = 1.196 m/s
Given that v1f = -v2f, we can substitute this into the equation:
-v2f + v2f = 1.196 m/s
Simplifying further:
2v2f = 1.196 m/s
Dividing both sides by 2:
v2f = 0.598 m/s
Since v1f = -v2f:
v1f = -0.598 m/s
Therefore, the final velocities of car #1 and car #2 after the collision are approximately -0.598 m/s and 0.598 m/s, respectively.