1) Mass m1 = 0.7 kg moves with constant velocity v1i = 2.0 m/s along the x-axis and collides with mass m2 = 1.8 kg, which moves with velocity v2i = 3.4 m/s, as sketched below. After the collision, m1 and m2 stick together. Find the angle, θ, between the x-axis and the direction of motion of the two masses after the collision. (Answer in degrees.)
θ =
To find the angle θ between the x-axis and the direction of motion of the two masses after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Step 1: Calculate the initial momentum of each mass.
The initial momentum, p1i, of mass m1 is given by:
p1i = m1 * v1i = (0.7 kg) * (2.0 m/s) = 1.4 kg·m/s
The initial momentum, p2i, of mass m2 is given by:
p2i = m2 * v2i = (1.8 kg) * (3.4 m/s) = 6.12 kg·m/s
Step 2: Calculate the final momentum of the combined masses.
Since m1 and m2 stick together after the collision, their final combined mass, Mf, is given by:
Mf = m1 + m2 = 0.7 kg + 1.8 kg = 2.5 kg
The final momentum, pfin, is given by:
pfin = Mf * vf = Mf * vf * cos(θ) (since the motion is along the x-axis)
Step 3: Apply conservation of momentum.
According to conservation of momentum, the initial momentum and the final momentum should be equal:
p1i + p2i = pfin
1.4 kg·m/s + 6.12 kg·m/s = 2.5 kg * vf * cos(θ)
7.52 kg·m/s = 2.5 kg * vf * cos(θ)
Step 4: Apply conservation of kinetic energy.
According to conservation of kinetic energy, the initial kinetic energy and the final kinetic energy should be equal:
(1/2) * m1 * v1i^2 + (1/2) * m2 * v2i^2 = (1/2) * Mf * vf^2
(1/2) * (0.7 kg) * (2.0 m/s)^2 + (1/2) * (1.8 kg) * (3.4 m/s)^2 = (1/2) * (2.5 kg) * vf^2
2.38 J + 10.932 J = 1.25 J * vf^2
13.312 J = 1.25 J * vf^2
Solving these two equations simultaneously will give you the value of vf and θ.