How Many Minutes?

When the two are aligned, the hour hand will be somewhat past the 8 mark and the minute hand will be somewhat past 2 mark. Call the number of minutes elapsed since 8 o'clock $X$ . The following proportionality will hold (because the angle shifts from the starting positions must be equal):

$\frac{X-10}{5} = \frac{X}{60}$

Solving for $X$ gives $X = 10.91$

The angular velocity of the hour-hand $\omega_h = \dfrac {360}{12\times 60} = \frac 12 ^\circ/\text{min.}$ The angular velocity of the minute-hand $\omega_m = \dfrac {360}{60} = 6^\circ/\text{min.}$ Let the time after 8:00 the two hands make a straight line for the first time be $t \text{ min.}$ In $t \text{ min.}$ the hour-hand would have moved $\omega_h t = \frac 12 t ^\circ$ . In the same time, the minute-hand would have to move $\frac 12 t ^\circ$ passed the second-hour mark or a total of $60^\circ + \frac 12 t ^\circ$ . Therefore, we have:

$\begin{aligned} 60^\circ + \frac 12 t ^\circ & = \omega_m t = 6 t \\ 120 & = 11 t \\ \implies t & = \frac {120}{11} \approx \boxed{10.91} \text{ min} \end{aligned}$

I wrote a general solution for this back in the day... for this one if M is the minutes past noon then

$40 + \frac{M}{12} = M + 30$

so

$10 = \frac{11}{12}M$

so

$M = \frac{120}{11}=10.91$

Vel of min hand be x then vel of min hand =6x and of hour hand =(1/2)x then apply relative velocity concept 6x - x/2 = 180 - 120

Take the clockwise direction to be the positive direction of angle measure.

Initial angle between hands = $120^{\circ}$

The hours hand moves by $30^{\circ}$ in $60$ minutes, therefore, by proportion,

Increment in angle of hours hand $= ( m / 60) * 30^{\circ} = m / 2^{\circ}$

The minutes hand moves by $30^{\circ}$ in $5$ minutes , therefore, applying this proportion,

Increment of angle of minutes hand $= (m / 5) * 30^{\circ} = 6 m^{\circ}$

Final Angle between hands $= 180^{\circ} =$ Initial Angle + Increment of minutes hands angle - Increment of hours hand angle

$180^{\circ} = 120^{\circ} + 6 m^{\circ} - \dfrac{1}{2} m^{\circ}$

Solving for m,

$m = \dfrac{120}{11} = 10.91$ minutes

Since there are 60 minutes in an hour, the minute hand moves 360 / 60 = 6 degree every minute. Hence the angle between the two hands is initially 20 * 6 = 120 degree. Since there are 12 * 60 minutes in 12 hours, the hour hand moves 360 / (12 * 60) = 0.5 degree every minute. Let x be the time in minutes until the two hands form a straight line: 120 + 6x - 0.5x = 180 Therefore 5.5 x equal 60 and so x = 10.90909