A 40 g marble moving at 2.3 m/s strikes a 29 g marble at rest. Assume the collision is perfectly elastic and the marbles collide head-on. What is the speed of the first marble immediately after the collision? What is the speed of the second marble immediately after the collision?
How do you set this up?
The first step is to use the conservation of momentum to solve for the velocities of the two marbles after the collision. Momentum is conserved, meaning that the total momentum before the collision must equal the total momentum after the collision.
The momentum of the first marble before the collision is 40 g x 2.3 m/s = 92 g m/s. The momentum of the second marble before the collision is 0 g m/s.
The total momentum before the collision is 92 g m/s.
The momentum of the first marble after the collision is 40 g x V1, where V1 is the velocity of the first marble after the collision. The momentum of the second marble after the collision is 29 g x V2, where V2 is the velocity of the second marble after the collision.
The total momentum after the collision is 40 g x V1 + 29 g x V2 = 92 g m/s.
Solving for V1 and V2, we get:
V1 = 2.3 m/s
V2 = 3.2 m/s
To set up this problem, we can use 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 a perfectly elastic collision.
The momentum of an object is defined as the product of its mass and velocity. So, we can calculate the momentum of each marble before the collision as:
Momentum of the first marble before the collision = mass of the first marble * velocity of the first marble
= 40 g * 2.3 m/s
Momentum of the second marble before the collision = mass of the second marble * velocity of the second marble
= 29 g * 0 m/s (since it's at rest)
After the collision, the total momentum should be conserved. The momentum of each marble after the collision can be calculated as:
Momentum of the first marble after the collision = mass of the first marble * velocity of the first marble after the collision
= 40 g * x (where x is the velocity of the first marble after the collision)
Momentum of the second marble after the collision = mass of the second marble * velocity of the second marble after the collision
= 29 g * y (where y is the velocity of the second marble after the collision)
Since the collision is perfectly elastic, the total momentum before the collision should be equal to the total momentum after the collision. This gives us the equation:
Momentum before collision = Momentum after collision
(40 g * 2.3 m/s) + (29 g * 0 m/s) = (40 g * x) + (29 g * y)
Now, we can solve this equation to find the values of x and y, which represent the velocities of the marbles after the collision.
Let's go ahead and calculate the speed of the first marble and the speed of the second marble immediately after the collision.
To solve this problem, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. Therefore, the momentum of an object with mass "m" and velocity "v" can be calculated as momentum = mass × velocity.
Let's calculate the initial momentum of the system (the total momentum before the collision):
Initial momentum = momentum of marble 1 + momentum of marble 2
The momentum of marble 1 (moving at 2.3 m/s) can be calculated as:
Momentum of marble 1 = mass of marble 1 × velocity of marble 1 = 40 g × 2.3 m/s
The momentum of marble 2 (at rest) can be calculated as:
Momentum of marble 2 = mass of marble 2 × velocity of marble 2 = 29 g × 0 m/s
Since marble 2 is at rest, its velocity is 0 m/s.
Therefore, the initial momentum of the system is:
Initial momentum = (40 g × 2.3 m/s) + (29 g × 0 m/s)
Now, let's calculate the final momentum of the system (the total momentum after the collision):
Final momentum = momentum of marble 1 (after collision) + momentum of marble 2 (after collision)
Since both marbles collide perfectly elastically, we know that the total momentum is conserved.
Now, we can set up the equation:
Initial momentum = Final momentum
Substituting the values calculated earlier, we get:
(40 g × 2.3 m/s) + (29 g × 0 m/s) = (mass of marble 1 (after collision) × velocity of marble 1 (after collision)) + (mass of marble 2 (after collision) × velocity of marble 2 (after collision))
Since we want to find the speeds of the marbles immediately after the collision, we can express their masses in grams and their velocities in m/s:
40 g × 2.3 m/s = m1 × v1 + 29 g × v2
From here, we can solve this equation to find the values of v1 (the speed of marble 1 immediately after the collision) and v2 (the speed of marble 2 immediately after the collision).