To find the velocity of the ozone molecule, we can break down the problem into components along the x and y axes.
Given data:
Mass of the oxygen molecule (O2) = 32 u
Velocity of the oxygen molecule (O2) in the x-direction = 580 m/s
Mass of the oxygen atom = 16 u
Velocity of the oxygen atom in the x-direction = 870 m/s
Angle between the oxygen atom's velocity vector and the x-axis = 27 degrees
To solve the problem, we need to find the final velocity of the ozone molecule. Since the oxygen molecule and oxygen atom stick together, the final mass is the sum of the two (32u + 16u = 48u).
Now, let's find the x and y components of the final velocity of the ozone molecule.
X-component of the final velocity:
The x-component of the oxygen molecule's velocity is already given as 580 m/s. For the oxygen atom's velocity, we need to calculate its x-component using the given angle and magnitude of the velocity vector.
Velocity of the oxygen atom in the x-direction = (870 m/s) * cos(27°)
Y-component of the final velocity:
The y-component of the oxygen molecule's velocity is zero since it is moving only in the x-direction. Similarly, for the oxygen atom's velocity, we need to calculate its y-component using the given angle and magnitude of the velocity vector.
Velocity of the oxygen atom in the y-direction = (870 m/s) * sin(27°)
Now, let's calculate the x and y components:
X-component:
Oxygen molecule's x-component = 580 m/s
Oxygen atom's x-component = (870 m/s) * cos(27°)
Y-component:
Oxygen molecule's y-component = 0
Oxygen atom's y-component = (870 m/s) * sin(27°)
Next, we can simply add up the x and y components to get the final velocity of the ozone molecule.
X-component of ozone's velocity = Oxygen molecule's x-component + Oxygen atom's x-component
Y-component of ozone's velocity = Oxygen molecule's y-component + Oxygen atom's y-component
So, the final velocity of the ozone molecule is expressed as a vector sum:
Velocity of the ozone molecule = (X-component of ozone's velocity) * i + (Y-component of ozone's velocity) * j
Substituting the calculated x and y components for the ozone molecule, we get:
Velocity of the ozone molecule = (580 m/s + (870 m/s) * cos(27°)) * i + ((870 m/s) * sin(27°)) * j
Thus, the final velocity of the ozone molecule is expressed in terms of the unit vectors i and j as:
Velocity of the ozone molecule = (580 + (870 * cos(27°))) * i + (870 * sin(27°)) * j
Therefore, the final answer is the expression: (580 + (870 * cos(27°))) * i + (870 * sin(27°)) * j, where i and j are unit vectors denoting the x and y directions, respectively.