This is a multi-step problem.  The first part to determine is how fast the stone was falling when it passed the window.  Knowing this will help us determine the height from which it was originally dropped.

Known:

Unknown:

Equation:

Using a kinematic equation, determine the speed the rock was moving when it first was at the top of the window.

Rearrange for the initial velocity.

Using this information as a final velocity it is possible to determine the height from which the stone originally fell. Additionally, since the object is assumed to fall from rest, the initial velocity is .

Knowns:

Unknowns:

Equations:

This means that the stone dropped  before hitting the window.

Knowns

Unknowns:

Time can be broken up as well.  There is a time that it takes the stone to land in the water below.  There is also a different time for the sound to reach the person’s ear.  This adds up to the total time provided in the problem.

Equation:

The stone travels with accelerated motion and the sound travels at a constant velocity.

Each step must be taken into account as the stone travels down the cliff and as the sound travels back. 

For the sound traveling the equation required is

Rearrange the equation to solve for the position as this is one of the factors that will connect both parts of this problem.

For the stone the best equation to be used is

Remember that the stone falls with an initial velocity of 0m/s so the equation can be simplified.

These equations are inverses of each other. (One is travelling from the top of the cliff to the ground and the other travels the other direction) So we can set them equal to the inverse of one another.

At this point, it is important to remember that there is a relationship between the time it takes for the rock to fall and the time for the sound to return to the person’s ear.

Therefore there is a relationship

Substitute this relationship into the equation above.

It is now possible to distribute on the right side.

This is a quadratic equation.  The next step is to rearrange, substitute values and solve the quadratic formula.

The two possible values are  and .  Time cannot be a negative value so the time for the rock to fall is .

Using this information, and returning to the original kinematic equation it is possible to find the height of the cliff.

The negative indicates that the stone fell  from its original position.  Therefore the height of the cliff is 

The only force accelerating the object is gravity since it was dropped, not thrown. Thus, to find out the speed of the object after some time, simply multiply the time the object has fallen by the acceleration of gravity. We will use . Then use the average velocity to calculate the distance the phone fell.

Final velocity after :

Distance the phone has fallen during the 9s of free fall:

Remember that the initial velocity of the phone is 0m/s.  This can be removed from the equation.

Gravity accelerates everything downward by 10m/s2. When the tomato is thrown upward with some velocity, gravity immediately begins to slowly reduce this velocity since the acceleration opposes the direction of the velocity.

At the top of the peak, the velocity of the tomato is .

Rearrange for time

In  the tomato will reach its maximum height. If you have ever thrown a ball upward you may have noticed how it appears to stop at the peak. We have just calculated the time it takes for that ball to appear to stop for a very small time and fall back down. Since our tomato must travel back to Earth, we double the time for its up and down motion (since they equal each other) to get  as the final answer.

The mass of an object is completely unrelated to its free-fall motion. The equation for the vertical motion for an object in freefall is:

Notice, there is no mention of mass anywhere in this equation. The only thing that affects the time an object takes to hit the ground is the acceleration due to gravity and the distance travelled. Since these objects travel the same distance and are affected by the same gravitational force, they will fall for the same amount of time and hit the ground together.

The question asks to point out the false statement. Everything on Earth is accelerated downwards by gravity, all the time, by . Think of gravity as having a negative sign. When the ball is thrown up, acceleration is working against the velocity slowing the ball down. Their signs are opposite. But when the ball is falling back down to Earth, the velocity and acceleration have the same sign. So velocity and acceleration will not always have the same sign.

If gravity could be said to have a negative sign since it pulls everything downward, then an upward velocity would have a positive (and opposing sign). At the top of the trajectory when the ball's upward velocity is finally overcome by gravity, the sign of the velocity becomes negative as it now points back down to Earth. So it is true velocity changes sign at the top.

Since the ball is caught at the same height, both the time up and time down are equal and it will be traveling at the same speed. Since gravity is the only force acting on it, the ball loses all its velocity on the way up and regains that exact amount by the time it reaches the height it started the journey. This also makes the time up equal to the time it falls. If the same force acts with same strength (gravity) the entire time, why would either of these change?

This is a multi-step problem and involves looking at the rocket at several points.

Point 0 - launch from the ground

Point 1 - when the fuel runs out

Point 2 - the height that the rocket reaches

Point 3 - when the rocket reaches the ground

Therefore all knowns and unknowns will be denoted by these different points.

To find the total time, it will be necessary to determine the time traveled under the acceleration, the time time traveled without acceleration to the peak and the time from the top of the peak to the ground.

Point 0 to Point 1

Equations:

The initial velocity is zero which simplifies the equation to

Rearrange to solve for time

It will also be important to know the speed that the rocket is traveling at when it is done accelerating.

Point 1 to Point 2

Equations:

Rearrange to solve for time

This time is the total time it takes to accelerate then reach the highest point of the peak.

It will also be important to know how high the rocket goes.

This is the total height that it takes the rocket to reach the highest point of the peak.

Point 2 to Point 3

Equations:

The initial velocity is zero which simplifies the equation to

Rearrange to solve for time

This is the total time for the rocket to go up and then come back down.

We are given the initial velocity, acceleration, and distance traveled. Using the equation below, we can solve for the time.  Remember that the initial velocity is 0m/s so it drops out of the equation.

The distance is negative, which makes since because the ball is traveling downward.  Also when taking the square root, only the positive value is needed as it is impossible to have negative time.

When examining vertical motion, the vertical velocity will always be zero at the highest point. At this point, the acceleration from gravity is working to change the motion of the ball from positive (upward) to negative (downward). This change is represented by the x-axis on a velocity versus time graph. As the ball changes direction, its velocity crosses the x-axis, momentarily becoming zero.