The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of 0.030 kg and is moving along the x axis with a velocity of +5.5 m/s. It makes a collision with puck B, which has a mass of 0.055 kg and is initially at rest. The collision is not head-on. After the collision, the two pucks fly apart with the angles shown in the drawing.
(a) Find the final speed of puck A.
(b) Find the final speed of puck B.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
(a) To find the final speed of puck A, we first need to calculate the initial momentum of puck A and puck B separately.
The initial momentum (p) of an object is given by its mass (m) multiplied by its velocity (v). Therefore, the initial momentum of puck A (PA) is
PA = mass of A × velocity of A
Given:
mass of A = 0.030 kg
velocity of A = +5.5 m/s (since the velocity is along the x-axis)
PA = 0.030 kg × 5.5 m/s
PA = 0.165 kg·m/s
After the collision, the two pucks fly apart with the angles shown in the drawing. This means they will have different velocities in the x and y directions.
To find the final velocity of puck A, we need to find its x and y components after the collision.
The x-component of the velocity of puck A (vAx) can be calculated using the conservation of momentum:
Initial momentum in x-direction = Final momentum in x-direction
PA × (initial velocity of A in x direction) = (mass of A) × (final velocity of A in x direction)
0.165 kg·m/s × 5.5 m/s = 0.030 kg × (final velocity of A in x direction)
(Since puck B is initially at rest, the final velocity of B in the x-direction is 0)
From the drawing, we can see that after the collision, the final velocity of A in the x-direction is equal to the initial velocity of A, which is 5.5 m/s.
Therefore, the x-component of the final velocity of puck A (vAx) = 5.5 m/s.
Next, let's calculate the y-component of the final velocity of puck A (vAy). Looking at the drawing, we can see that the vertical components of the velocities of both pucks are equal in magnitude and opposite in direction, so they cancel each other out.
Therefore, the y-component of the final velocity of puck A (vAy) = 0 m/s.
Now, we can use the Pythagorean theorem to find the magnitude of the final velocity of puck A (vA):
vA = √(vAx^2 + vAy^2)
vA = √(5.5 m/s)^2 + (0 m/s)^2
vA = √(30.25 m^2/s^2)
vA ≈ 5.5 m/s
Therefore, the final speed of puck A is approximately 5.5 m/s.
(b) To find the final speed of puck B, we use the principles of conservation of momentum and conservation of kinetic energy.
Since the horizontal components of the velocities of both pucks cancel out after the collision, we only need to consider the vertical components.
The y-component of the final velocity of puck A (vAy) cancels out with the y-component of the final velocity of puck B (vBy) since they are equal in magnitude and opposite in direction.
We can write the equation for conservation of momentum in the y-direction:
Initial momentum in y-direction = Final momentum in y-direction
0 = (mass of A) × (final velocity of A in y direction) + (mass of B) × (final velocity of B in y direction)
Let's denote the final velocity of puck B (in the y-direction) as vb:
0 = 0.030 kg × 0 m/s + 0.055 kg × vb
vb = 0 m/s
Therefore, the final speed of puck B is 0 m/s.