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 principle of conservation of momentum.
1. First, calculate the initial momentum (p_initial) of the system before the collision. Momentum is given by the product of mass and velocity:
p1_initial = m1 * v1i
p2_initial = m2 * v2i
p_initial = p1_initial + p2_initial
2. Since the masses stick together after the collision, they move with a common final velocity (vf).
Let's assume the angle between the x-axis and the final velocity of the masses is θ.
The x-component of the final momentum (p_final_x) of the system is given by:
p_final_x = (m1 + m2) * vf * cos(θ)
The y-component of the final momentum (p_final_y) of the system is given by:
p_final_y = (m1 + m2) * vf * sin(θ)
3. Apply the conservation of momentum principle, which states that the total initial momentum of a system is equal to the total final momentum of the system:
p_initial = p_final_x
m1 * v1i + m2 * v2i = (m1 + m2) * vf * cos(θ)
4. Also, there is no momentum in the y-direction before and after the collision, so:
p_initial_y = p_final_y
0 = (m1 + m2) * vf * sin(θ)
5. Solve the first equation for vf:
vf = (m1 * v1i + m2 * v2i) / (m1 + m2)
6. Plug the value of vf into the second equation:
0 = (m1 + m2) * ((m1 * v1i + m2 * v2i) / (m1 + m2)) * sin(θ)
Simplify:
0 = (m1 * v1i + m2 * v2i) * sin(θ)
7. Solve the equation for θ:
sin(θ) = 0
Since sin(θ) = 0, the angle θ is 0 degrees.
Therefore, the angle, θ, between the x-axis and the direction of motion of the two masses after the collision is 0 degrees.