A 26.3 g marble sliding to the right at 21.0 cm/s overtakes and collides elastically with a
13.3 g marble moving in the same direction at 12.2 cm/s. After the collision, the 13.3 g marble moves to the right at 23.7 cm/s.
Find the velocity of the 26.3 g marble after the collision.
Answer in units of cm/s
Conservation of momentum
Intial momentum=final momentum
26.3*21+13.3*12.2=13.3*23.7+26.2V
solve for V
To find the velocity of the 26.3 g marble after the collision, we can make use of the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Mathematically, momentum (p) is calculated as:
p = mass (m) × velocity (v)
Let's denote the velocity of the 26.3 g marble after the collision as v1 and the velocity of the 13.3 g marble after the collision as v2. The total momentum before the collision (p_total_initial) is the sum of the individual momenta of the two marbles:
p_total_initial = (mass 1 × velocity 1) + (mass 2 × velocity 2)
Substituting the given values:
p_total_initial = (26.3 g × 21.0 cm/s) + (13.3 g × 12.2 cm/s)
Now, according to the conservation of momentum:
p_total_initial = p_total_final
Since the collision is elastic, the momentum of each marble is conserved. Therefore, the total momentum after the collision (p_total_final) is also the sum of the individual momenta of the marbles after the collision:
p_total_final = (26.3 g × v1) + (13.3 g × v2)
Since the 13.3 g marble moves to the right at 23.7 cm/s after the collision (v2 = 23.7 cm/s), we can substitute this value into the equation:
p_total_final = (26.3 g × v1) + (13.3 g × 23.7 cm/s)
Now, we equate p_total_initial and p_total_final:
(26.3 g × 21.0 cm/s) + (13.3 g × 12.2 cm/s) = (26.3 g × v1) + (13.3 g × 23.7 cm/s)
Simplifying the equation:
549.3 g·cm/s + 162.26 g·cm/s = (26.3 g × v1) + (315.21 g·cm/s)
711.56 g·cm/s = (26.3 g × v1) + (315.21 g·cm/s)
Now, we can solve for v1:
v1 = (711.56 g·cm/s - 315.21 g·cm/s) / 26.3 g
Calculating the values:
v1 = 8.91 cm/s
Therefore, the velocity of the 26.3 g marble after the collision is 8.91 cm/s.