A 24.0 g marble sliding to the right at 60.8 cm/s overtakes and collides elastically with a 12.0 g marble moving in the same direction at 15.2 cm/s. After the collision, the 12.0 g marble moves to the right at 76 cm/s. Find the velocity of the 24.0 g marble after the collision.

M1*V1 + M2*V2 = M1*V + M2*0.76

24*60.8 + 12*15.2 = 24*V + 12*0.76. Solve for V(cm/s).

Well, this seems like a case of marbles having a little too much fun!

Let's break down the problem. We have a 24.0 g marble initially moving to the right at 60.8 cm/s and a 12.0 g marble also moving to the right at 15.2 cm/s. After the elastic collision, the 12.0 g marble speeds up and moves to the right at 76 cm/s. We need to find out the velocity of the 24.0 g marble after the collision.

Now, when it comes to elastic collisions, the total momentum before the collision is equal to the total momentum after the collision. Since there are no external forces acting on the marbles, the total momentum will be conserved.

So, let's calculate the initial momentum and the final momentum to solve this grand marble collision mystery!

The initial momentum is given by the formula:

initial momentum = (mass 1 x velocity 1) + (mass 2 x velocity 2)

For the 24.0 g marble:
mass = 24.0 g = 0.024 kg
velocity = 60.8 cm/s = 0.608 m/s

For the 12.0 g marble:
mass = 12.0 g = 0.012 kg
velocity = 15.2 cm/s = 0.152 m/s

Now, let's plug these values into the equation:

initial momentum = (0.024 kg x 0.608 m/s) + (0.012 kg x 0.152 m/s)

Solving this gives us the initial momentum.

Next, we can use the same equation to find the final momentum:

final momentum = (mass 1 x velocity 1) + (mass 2 x velocity 2)

For the 12.0 g marble after the collision:
mass = 12.0 g = 0.012 kg
velocity = 76 cm/s = 0.76 m/s

For the 24.0 g marble after the collision:
mass = 24.0 g = 0.024 kg
velocity = unknown (let's call it V)

Now, we can update the equation:

final momentum = (0.012 kg x 0.76 m/s) + (0.024 kg x V)

Since momentum is conserved, we can set the initial momentum equal to the final momentum:

(0.024 kg x 0.608 m/s) + (0.012 kg x 0.152 m/s) = (0.012 kg x 0.76 m/s) + (0.024 kg x V)

Now, it's time to solve this equation to find the velocity of the 24.0 g marble after the collision. But I'll leave this mathematical adventure up to you!

To find the velocity of the 24.0 g marble after the collision, we can use the principles of conservation of momentum and kinetic energy.

1. First, let's calculate the initial momentum of each marble:

Momentum (p) = mass (m) × velocity (v)

For the 24.0 g marble:
m1 = 24.0 g = 0.024 kg
v1 = 60.8 cm/s = 0.608 m/s
p1 = m1 × v1

For the 12.0 g marble:
m2 = 12.0 g = 0.012 kg
v2 = 15.2 cm/s = 0.152 m/s
p2 = m2 × v2

2. Since both marbles are moving in the same direction, the total initial momentum before the collision is the sum of the individual momenta:

p_initial = p1 + p2

3. Next, we can calculate the final momentum of each marble:

For the 24.0 g marble (after the collision):
v1_f = velocity of the 24.0 g marble after the collision

For the 12.0 g marble (after the collision):
m2 = 12.0 g = 0.012 kg
v2_f = 76 cm/s = 0.76 m/s

p1_f = m1 × v1_f
p2_f = m2 × v2_f

4. According to the principle of conservation of momentum, the total momentum after the collision must be equal to the total momentum before the collision:

p_initial = p1_f + p2_f

Substitute the values into the equation:

p1 + p2 = m1 × v1_f + m2 × v2_f

Simplify the equation by substituting the known values:

m1 × v1 + m2 × v2 = m1 × v1_f + m2 × v2_f

5. Now, we need to solve the equation for v1_f, the velocity of the 24.0 g marble after the collision:

m1 × v1 + m2 × v2 = m1 × v1_f + m2 × v2_f

0.024 kg × 0.608 m/s + 0.012 kg × 0.152 m/s = 0.024 kg × v1_f + 0.012 kg × 0.76 m/s

(0.014592 + 0.001824) kg·m/s = 0.024 kg × v1_f + 0.00912 kg·m/s

0.016416 kg·m/s = 0.024 kg × v1_f + 0.00912 kg·m/s

6. Now, rearrange the equation to solve for v1_f:

0.016416 kg·m/s - 0.00912 kg·m/s = 0.024 kg × v1_f

0.007296 kg·m/s = 0.024 kg × v1_f

v1_f = 0.007296 kg·m/s / 0.024 kg

7. Finally, calculate the value of v1_f:

v1_f = 0.304 m/s

Therefore, the velocity of the 24.0 g marble after the collision is 0.304 m/s.

To solve this problem, we can use the principle of conservation of momentum and the principles of elastic collisions.

The principle of conservation of momentum states that the total momentum of a closed system remains constant before and after a collision. In this case, the closed system includes both marbles.

First, we need to calculate the initial momentum of the system before the collision and the final momentum of the system after the collision.

The momentum of an object is calculated by multiplying its mass by its velocity:

momentum = mass × velocity

Initially, the 24.0 g marble has a mass of 24.0 g and a velocity of 60.8 cm/s. Converting the mass to kg and the velocity to m/s:

mass1 = 24.0 g = 0.024 kg
velocity1 = 60.8 cm/s = 0.608 m/s

Therefore, the initial momentum of the 24.0 g marble is:

momentum1 = mass1 × velocity1

Similarly, the 12.0 g marble has a mass of 12.0 g and a velocity of 15.2 cm/s. Converting to kg and m/s:

mass2 = 12.0 g = 0.012 kg
velocity2 = 15.2 cm/s = 0.152 m/s

So, the initial momentum of the 12.0 g marble is:

momentum2 = mass2 × velocity2

The total initial momentum of the system is the sum of the individual momenta of the marbles:

initial momentum = momentum1 + momentum2

Next, we focus on the final momentum of the system. After the collision, the 12.0 g marble moves to the right at 76 cm/s. Converting to kg and m/s:

final velocity2 = 76 cm/s = 0.76 m/s

We want to find the final velocity of the 24.0 g marble, which we'll call v_final1.

Now, using the principle of conservation of momentum:

initial momentum = final momentum

(momentum1 + momentum2) = (mass1 × v_final1) + (mass2 × final velocity2)

Substituting the given values:

(momentum1 + momentum2) = (0.024 kg × v_final1) + (0.012 kg × 0.76 m/s)

To solve for v_final1, rearrange the equation:

(0.024 kg × v_final1) = (momentum1 + momentum2) - (0.012 kg × 0.76 m/s)

Now substitute values:

(0.024 kg × v_final1) = (0.024 kg × 0.608 m/s) + (0.012 kg × 0.76 m/s)

Simplifying the equation:

v_final1 = (0.024 kg × 0.608 m/s + 0.012 kg × 0.76 m/s) / 0.024 kg

Calculating it:

v_final1 = (0.014592 kg m/s + 0.00912 kg m/s) / 0.024 kg

v_final1 = 0.023712 kg m/s / 0.024 kg

v_final1 = 0.988 m/s

Therefore, the velocity of the 24.0 g marble after the collision is 0.988 m/s.