The distance traveled is directly related to the square of the time traveled.  Therefore time if time is doubled, the value will be squared and therefore four times as great.  The distance traveled will be 4 times as far.

The can will have the same acceleration both up and down the hill since there is no friction.  Friction or other forces would cause a change in acceleration.  But since there is no friction, the can will travel up the hill, slow down and then accelerate back down the hill.

Knowns:

Unknown:

Equation:

The first thing is to determine the initial velocity of the runner before the runner accelerates for the final portion of the race.  Since the runner is traveling at a constant velocity

Next, convert the time during the constant speed portion to seconds.

Determine the amount of distance traveled while at a constant speed.

Use these values to determine the velocity of the runner.

Next determine the final velocity needed for the runner to finish out the race in the remaining time.

 left in race

Finally use the kinematic equations to calculate the time needed to get to this velocity.

Rearrange for time

Explanation:

We are given the initial velocity, time, and final velocity (zero because the ball stops). Using these values and the appropriate motion equation, we can solve for the acceleration.

Acceleration is given by the change in velocity over time:

We can use our values to solve for the acceleration.

There is a direct relationship between the final velocity and the time traveled.  Therefore if the time is doubled, the velocity would double as well.

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.