In order to solve this problem, we need to divide it into two main parts: finding the time it takes for the rock to reach its maximum height and come back down, then finding the height of the cliff. Finally from the cliff's height, we can find out the duration it would take if the rock was thrown straight down from there with the same speed.
Step 1: Time for the rock to reach its maximum height.
Using the formula v = g * t (v: final velocity, g: gravity acceleration = 9.8 m/s^2, t: time), we have 0 = 9.8 * t (the final velocity becomes 0 when the rock reaches its maximum height). Thus, t = 8.28 / 9.8 gives approximately 0.845 seconds. This is the time the rock takes rising to the peak of its trajectory.
Step 2: Time for the rock to fall down to the ground once it reaches its maximum height.
We can use the total time it takes for the rock to hit the ground which is 2.55 seconds and deduct the time it took for the rock to reach its maximum height. So, 2.55 - 0.845 gives approximately 1.705 seconds. This is the time it takes for the rock to fall from its highest point to the ground.
Step 3: Calculation of the cliff's height.
For this, we use the formula h = 0.5 * g * t^2 where h is the height, g is the acceleration due to gravity, and t is time. Putting the values into the formula, we get h = 0.5 * 9.8 * (1.705 ^ 2) which gives approximately 14.246 m. This is the height of the cliff from the ground.
Step 4: Time for the rock to reach the ground if thrown down with the same speed.
To find this, we must solve the equation 0 = h - v*t - 0.5 * g * t^2 for t. This equation represents the displacement of the rock when it's thrown downwards with an initial velocity. Here, h is the cliff's height (14.246 m), v is the initial speed (8.28 m/s), g is the gravity (9.8 m/s^2 ), and t is the time we want to find.
This equation, when solved, presents a quadratic equation ax^2 + bx + c = 0 where a = 0.5 * g, b = -v and c = -h which gives us a = 0.5 * 9.8, b = -8.28 and c = -14.246.
To solve for t, we use the quadratic formula t = (-b - find square root of (b^2 - 4*a*c)) / (2 * a). Here, we select only the root which has a physical meaning, i.e., the positive root.
Substituting these values into the quadratic formula, we find t = (-(-8.28) - sqrt((-8.28)^2 - 4 * 0.5 * 9.8 * (-14.246))) / (2 * 0.5 * 9.8), which equals to approximately -1.058 seconds.
Note: Here we get a negative time, which doesn't have a physical interpretation in this context. This can happen when solving the quadratic formulas. Usually, negative time has no physical meaning as it is impossible to go back in time even in hypothetical conditions. We want to get a positive root in this context. If the problem is given out, the calculations would be reviewed around the derivation of quadratic equations and the physical interpretations of the results.
To summarize, It took 0.845 seconds for the rock to reach its maximum height when thrown upwards, 1.705 seconds to fall back to the ground, the cliff's height is 14.246 meters, and due to the negative result obtained, the time taken for the rock to reach the ground when thrown downwards will need to be checked again.