Step 1: Know what the slope of a v-t graph means.
On a velocity time graph, the slope of the line tells us the acceleration. This is because acceleration is how fast velocity changes with time, which is exactly rise over run on this graph.
Step 2: Turn an angle into a slope.
If a line makes an angle $\theta$ with the time axis, its slope is $\tan\theta$. So the acceleration of each object equals the tangent of its angle.
Step 3: Write acceleration for the first object.
The first line makes $30^{\circ}$ with the time axis: \[ a_1 = \tan 30^{\circ} = \frac{1}{\sqrt{3}} \]
Step 4: Write acceleration for the second object.
The second line makes $45^{\circ}$ with the time axis: \[ a_2 = \tan 45^{\circ} = 1 \]
Step 5: Form the ratio.
Divide the first acceleration by the second: \[ \frac{a_1}{a_2} = \frac{1/\sqrt{3}}{1} = \frac{1}{\sqrt{3}} \]
Step 6: State the answer.
So the accelerations are in the ratio: \[ \boxed{a_1 : a_2 = 1 : \sqrt{3}} \]