A 24.9 g marble sliding to the right at 22.0
cm/s overtakes and collides elastically with a
13.2 g marble moving in the same direction
at 11.8 cm/s. After the collision, the 13.2 g
marble moves to the right at 25.0 cm/s.
Find the velocity of the 24.9 g marble after
the collision.
Answer in units of cm/s.
M1*V1 + M2*V2 = M1*V + M2*25.
24.9*22 + 13.2*11.8 = 24.9V + 13.2*25.
V = ?
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
1. First, we need to find the initial momentum of the two marbles before the collision. Momentum is defined as mass times velocity.
The momentum of the 24.9 g marble (m1) is given by:
m1 = mass1 * velocity1 = 24.9 g * 22.0 cm/s
The momentum of the 13.2 g marble (m2) is given by:
m2 = mass2 * velocity2 = 13.2 g * 11.8 cm/s
2. Next, we can calculate the total momentum before the collision.
Total momentum before collision = m1 + m2
3. According to the principle of conservation of momentum, the total momentum after the collision will be equal to the total momentum before the collision.
Total momentum after collision = Total momentum before collision
4. Let's assume the final velocity of the 24.9 g marble (v1) and the final velocity of the 13.2 g marble (v2) after the collision.
The momentum of the 24.9 g marble after the collision (m1') is given by:
m1' = mass1 * v1
The momentum of the 13.2 g marble after the collision (m2') is given by:
m2' = mass2 * v2
5. We can set up two equations using the conservation of momentum principle:
m1 + m2 = m1' + m2' (equation 1)
m1' + m2' = m1 * v1 + m2 * v2 (equation 2)
6. Now, we also have the information that the final velocity of the 13.2 g marble after the collision is 25.0 cm/s.
Using this information, we can substitute the values into equation 2:
m1' + m2' = m1 * v1 + m2 * 25.0 cm/s
7. We can solve equation 2 for v1:
v1 = (m1' + m2' - m2 * 25.0 cm/s) / m1
8. Finally, substitute the known values into the equation and calculate v1:
v1 = (m1' + m2' - m2 * 25.0 cm/s) / m1
Now you can solve for v1 to find the velocity of the 24.9 g marble after the collision.
To find the velocity of the 24.9 g marble after the collision, we can use the principles of conservation of momentum and kinetic energy.
Conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
The equation for momentum is given by:
momentum = mass x velocity
We are given the masses and velocities of both marbles before and after the collision:
m1 = 24.9 g (mass of the 24.9 g marble before the collision)
v1i = 22.0 cm/s (velocity of the 24.9 g marble before the collision)
m2 = 13.2 g (mass of the 13.2 g marble before the collision)
v2i = 11.8 cm/s (velocity of the 13.2 g marble before the collision)
v2f = 25.0 cm/s (velocity of the 13.2 g marble after the collision)
First, we need to calculate the initial momentum of both marbles before the collision.
p1i = m1 * v1i
p1i = 24.9 g * 22.0 cm/s
p2i = m2 * v2i
p2i = 13.2 g * 11.8 cm/s
Next, we can calculate the total momentum before the collision.
p_total_i = p1i + p2i
Now, we need to calculate the final momentum of both marbles after the collision. Since the collision is elastic, the total kinetic energy is conserved.
The equation for kinetic energy is given by:
kinetic energy = 0.5 x mass x velocity^2
We can calculate the initial kinetic energy of both marbles before the collision.
KE1i = 0.5 x m1 x v1i^2
KE1i = 0.5 x 24.9 g x (22.0 cm/s)^2
KE2i = 0.5 x m2 x v2i^2
KE2i = 0.5 x 13.2 g x (11.8 cm/s)^2
Next, we calculate the final kinetic energy of both marbles after the collision.
KE2f = 0.5 x m2 x v2f^2
KE2f = 0.5 x 13.2 g x (25.0 cm/s)^2
Since the total kinetic energy is conserved, we have:
KE_total_i = KE1i + KE2i
KE_total_f = KE1f + KE2f
Given that KE2f = KE2i (since the 13.2 g marble is the only one that experiences a change in velocity), we can re-write the equation as:
KE_total_f = KE1f + KE2i
Now, we can solve for KE_total_f:
KE_total_f = 0.5 x m1 x v1f^2 + 0.5 x m2 x v2i^2
Since kinetic energy is given by the equation:
kinetic energy = 0.5 x mass x velocity^2
We can equate KE_total_f to the final momentum of the system.
KE_total_f = 0.5 x (m1 + m2) x v_f^2
Setting the two equations equal to each other:
0.5 x m1 x v1f^2 + 0.5 x m2 x v2i^2 = 0.5 x (m1 + m2) x v_f^2
Now we can solve for vf by rearranging the equation:
v_f^2 = (m1 x v1f^2 + m2 x v2i^2) / (m1 + m2)
Finally, we take the square root of both sides to find the velocity of the 24.9 g marble after the collision:
v_f = sqrt((m1 x v1f^2 + m2 x v2i^2) / (m1 + m2))
Plugging in the given values:
m1 = 24.9 g
v1i = 22.0 cm/s
m2 = 13.2 g
v2i = 11.8 cm/s
v2f = 25.0 cm/s
v_f = sqrt((24.9 g x (22.0 cm/s)^2 + 13.2 g x (11.8 cm/s)^2) / (24.9 g + 13.2 g))
Now, we can calculate the value using a calculator:
v_f ≈ 18.2 cm/s
Therefore, the velocity of the 24.9 g marble after the collision is approximately 18.2 cm/s.