To solve this problem, we use the equation of motion:

\(v^2 = u^2 + 2as\)

where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the distance.

1. (a) For the distance \(d_1\) between points \(A\) and \(B\):

   Initial speed \(u = 5 \text{ m s}^{-1}\), final speed \(v = 7 \text{ m s}^{-1}\).

   \(7^2 = 5^2 + 2ad_1\)

   \(49 = 25 + 2ad_1\)

   \(24 = 2ad_1\)

   \(12 = ad_1\)

2. (b) For the distance \(d_2\) between points \(B\) and \(C\):

   Initial speed \(u = 7 \text{ m s}^{-1}\), final speed \(v = 8 \text{ m s}^{-1}\).

   \(8^2 = 7^2 + 2ad_2\)

   \(64 = 49 + 2ad_2\)

   \(15 = 2ad_2\)

   \(7.5 = ad_2\)

Now, to find \(d_1\) in terms of \(d_2\):

From \(ad_1 = 12\) and \(ad_2 = 7.5\), we have:

\(\frac{ad_1}{ad_2} = \frac{12}{7.5}\)

\(\frac{d_1}{d_2} = \frac{24}{15}\)

\(\frac{d_1}{d_2} = 1.6\)

Thus, \(d_1 = 1.6d_2\).