A spring loaded 100g dart gun is affixed at rest on a 150g frictionless cart. It fires a 25g dart with a velocity of 30 m/s. What is the final momentum of the system?
I'm confused as to whether or not I should use conservation of momentum or center of mass, and I'm not really sure what to do if it is center of mass...
Thank you!
ZERO !!
The final momentum =initial momentum
Which IS ZERO
if you want the speed backwards of the cart then
250 v + 25 * 30 = 0
so the momentum of the cart is 250 v
By the way the center of mass of the SYSTEM (not the cart) also does not move because there is no external force on the SYSTEM.
The car moves back as the dart moves forward to keep the momentum and center of gravity the same before and after.
Thank you!
Well, in this case, you can apply the conservation of momentum principle. According to this principle, the total momentum of an isolated system is conserved before and after any interaction. So, the initial momentum of the system will be equal to the final momentum.
Let's break it down. The initial momentum of the system is given by:
Initial momentum = (mass of dart gun * velocity of dart gun) + (mass of cart * velocity of cart)
= (0.1 kg * 0 m/s) + (0.15 kg * 0 m/s) (since both the dart gun and cart are at rest initially)
= 0 kg m/s
Now, after the dart is fired, the final momentum of the system is given by:
Final momentum = (mass of dart gun * final velocity of dart gun) + (mass of cart * final velocity of cart)
= (0.1 kg * -30 m/s) + (0.15 kg * final velocity of cart)
Since the dart gun experiences an equal and opposite impulse due to firing the dart, its final velocity becomes -30 m/s (opposite direction). The final velocity of the cart, however, is unknown.
So, the final momentum of the system is:
Final momentum = -3 kg m/s + (0.15 kg * final velocity of cart)
Now, if you know the final velocity of the cart, you can substitute it into the equation to find the final momentum of the system.
To find the final momentum of the system, you can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision or any other interaction.
In this case, the system consists of the dart gun and the cart. Before firing the dart, the system is at rest, so the initial momentum is zero.
After the dart is fired, the system will experience an interaction that causes the cart to move in the opposite direction to conserve momentum. The dart itself will continue moving forward due to the force exerted by the spring.
To find the final momentum of the system, you need to consider the momentum of both the cart and the dart. The momentum of an object can be calculated by multiplying its mass with its velocity.
Given:
- Mass of the cart (m1) = 150g = 0.15 kg
- Mass of the dart (m2) = 25g = 0.025 kg
- Initial velocity of the cart and dart (u) = 0 m/s (at rest)
- Final velocity of the dart (v2) = 30 m/s (forward)
Calculating the final velocity of the cart (v1):
By using the conservation of momentum principle: m1u + m2u = m1v1 + m2v2
Since the initial velocities are zero, the equation simplifies to: 0 = m1v1 + m2v2
Rearranging the equation: -m2v2 = m1v1
Substituting the given values: -0.025 kg * 30 m/s = 0.15 kg * v1
Simplifying the equation: -0.75 kg*m/s = 0.15 kg * v1
Dividing both sides of the equation by 0.15 kg: -5 m/s = v1
So, the final velocity of the cart (v1) is -5 m/s (backward).
Now we can calculate the final momentum of the system:
The momentum of the cart (p1) is given by: p1 = m1 * v1 = 0.15 kg * (-5 m/s) = -0.75 kg*m/s (backward)
The momentum of the dart (p2) is given by: p2 = m2 * v2 = 0.025 kg * 30 m/s = 0.75 kg*m/s (forward)
The final momentum of the system is the sum of the individual momenta:
Final momentum = p1 + p2 = -0.75 kg*m/s + 0.75 kg*m/s = 0 kg*m/s
Therefore, the final momentum of the system is 0 kg*m/s.
Note: The center of mass is a concept used to locate the average position of mass in a system. It is not directly related to finding the final momentum in this context. Conservation of momentum is used instead to solve for the final momentum of the system.