Mass m1 = 0.5 kg moves with constant velocity v1i = 2.5 m/s along the x-axis and collides with mass m2 = 2.6 kg, which moves with velocity v2i = 3.1 m/s, as sketched below(completely vertical). After the collision, m1 and m2 stick together.
a.) Find the angle, θ, between the x-axis and the direction of motion of the two masses after the collision. (answer in degrees)
b.)What is the magnitude of the final momentum?
c.)What is the magnitude of the final velocity?
A 20.0 kg box slides down a 12.0 m long incline at an angle of 3.0 degrees with the horizontal. A force of 50.0 N is applied to the box to try to pull it up the incline. The applied force makes an angle of 0.00 degrees to the incline. If the incline has a coefficient of kinetic friction of 0.100, then the increase in the kinetic energy of the box is:
To solve this problem, we can apply the principles of conservation of momentum and conservation of kinetic energy.
First, let's find the velocity of the two masses after the collision.
Step 1: Find the initial momentum of each mass:
p1i = m1 * v1i
p1i = 0.5 kg * 2.5 m/s
p1i = 1.25 kg·m/s
p2i = m2 * v2i
p2i = 2.6 kg * 3.1 m/s
p2i = 8.06 kg·m/s
Step 2: Apply conservation of momentum:
Since the masses stick together after the collision, their combined momentum is conserved.
p_tot = p1f + p2f
Where p_tot is the total momentum before the collision, and p1f and p2f are the final momenta of mass m1 and m2.
p_tot = 1.25 kg·m/s + 8.06 kg·m/s
p_tot = 9.31 kg·m/s
Step 3: Find the final velocity of the combined masses:
After the collision, the combined masses move together with a final velocity, v_f.
p_tot = (m1 + m2) * v_f
9.31 kg·m/s = (0.5 kg + 2.6 kg) * v_f
9.31 kg·m/s = 3.1 kg * v_f
v_f = 9.31 kg·m/s / 3.1 kg
v_f = 3 m/s
(a) To find the angle, θ, between the x-axis and the direction of motion of the two masses after the collision, we can use trigonometry. Since the masses move together in a straight line after the collision, their velocity vector forms an angle with the x-axis.
The tangent of the angle can be found using the y and x components of the velocity vector:
tan(θ) = (Δy) / (Δx)
tan(θ) = 0 / 3
tan(θ) = 0
θ = arctan(0)
θ = 0°
(b) The magnitude of the final momentum is the absolute value of the total momentum after the collision:
|p_tot| = |9.31 kg·m/s| ≈ 9.31 kg·m/s
(c) The magnitude of the final velocity is the absolute value of the combined velocity of the two masses after the collision:
|v_f| = |3 m/s| ≈ 3 m/s
So, the answers to the questions are:
(a) θ = 0°
(b) The magnitude of the final momentum is 9.31 kg·m/s.
(c) The magnitude of the final velocity is 3 m/s.