To find the correct answer, let's go over the steps of the problem and recheck the calculations.
First, let's calculate the final velocities of Betty and Sally after the collision.
Given:
mBetty = 55 kg
vBetty = 22.0 km/h = 22.0 * (1000/3600) m/s (convert km/h to m/s)
mSally = 45 kg
vSally = 28.0 km/h = 28.0 * (1000/3600) m/s (convert km/h to m/s)
We can find the components of their velocities in the x and y directions:
vxBetty = vBetty * cos(76.0°) (since 76.0° counterclockwise from the x-direction)
vyBetty = vBetty * sin(76.0°)
vxSally = vSally * cos(12.0°) (since 12.0° from the x-axis)
vySally = vSally * sin(12.0°)
Next, let's use the law of conservation of momentum to find the final velocities:
Initial momentum in the x direction = Final momentum in the x direction
mBetty * vBetty = mBetty * vxBetty + mSally * vxSally
From this equation, we can solve for vxBetty.
Similarly, the initial momentum in the y direction equals the final momentum in the y direction, so we have:
mBetty * 0 + mSally * vSally = mBetty * vyBetty + mSally * vySally
From this equation, we can solve for vyBetty.
Now that we have the components of Betty's velocity, we can calculate her final overall velocity:
vBetty (final) = sqrt(vxBetty^2 + vyBetty^2)
Finally, we can calculate the final kinetic energy for both Betty and Sally using the formula:
K = 1/2 * m * v^2
where m is the mass and v is the velocity.
With these steps, go ahead and recalculate the final velocities and kinetic energies for Betty and Sally and compare them to the book's answer.