Final answer:
After the collision, the two suction-cup darts have a combined speed of 0 m/s because the total momentum is conserved, and the darts have equal and opposite velocities with mass ratios of 1:2, resulting in them coming to rest after sticking together.
Step-by-step explanation:
To determine the speed of two suction-cup darts that collide head-on and stick together, we can apply the principle of conservation of linear momentum. Since there are no external forces acting on the system, the total momentum before the collision must be equal to the total momentum after the collision.
Let's denote the mass of the first dart as m1 = 1.13 grams = 0.00113 kg (since we need the mass in kilograms to use the SI units correctly), which moves to the left with a velocity of v1 = -24.0 m/s. The second dart, with twice the mass, will be m2 = 2 * 1.13 grams = 2.26 grams = 0.00226 kg, moving to the right with a velocity of v2 = 24.0 m/s. The negative sign for v1 indicates that it is moving in the opposite direction to v2.
Using the conservation of momentum:
m1 * v1 + m2 * v2 = (m1 + m2) * v_final
Plugging in the values:
(0.00113 kg)(-24.0 m/s) + (0.00226 kg)(24.0 m/s) = (0.00113 kg + 0.00226 kg) * v_final
-0.02712 kg*m/s + 0.05424 kg*m/s = 0.00339 kg * v_final
v_final = (0.02712 kg*m/s) / 0.00339 kg
Calculating v_final, we find that the final velocity is 0 m/s, meaning the darts come to rest after the collision, sticking together due to the conservation of momentum and the equal and opposite velocities of the same magnitude.