The problem gives us the distance, the acceleration due to gravity, and implies that the initial velocity of the picture is zero, as it starts at rest.

We can find the final velocity using the appropriate motion equation:

We can use our values to solve for the final velocity. Keep in mind that the displacement will be negative because the ball is traveling in the downward direction!

The square root of a term can be either positive or negative, depending on the direction. Since velocity is a vector, and the painting is falling downward, our final answer will be negative: .

The relationship between velocity, distance, and time is:

We can multiply both sides by the time, then divide both sides by the velocity, to isolate the variable for time.

The quotient of distance and velocity will give us the time that the object was in motion.

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.

The woman needs to swim , or . She normally swims at , but has a current opposing her. The effect will make her relative velocity equal to her normal velocity, minus the current against her.

We know that velocity is a change in distance divided by a change in time. Now that we have her relative velocity and the distance traveled, we can isolate the time variable and solve for her time.

To solve this problem we can simply examine each biker separately and see how long it would take them to reach the destination. Let's begin with the recreational biker. The recreational biker needs to ride  at . Using the definition of velocity, we can find his final time.

The professional has a distance of , plus the  that he's behind.

We know that the velocity of the professional biker is . Using this velocity and his total distance, we can find the time that it takes him to reach the end of the path.

The time of the professional biker is less than that of the recreational biker, meaning that the professional will finish first. Now we need to find the distance between the two bikers at this point. Use the recreational biker's velocity and the time difference between the two bikers to solve for the distance that the recreational biker has left on the path.

The recreational biker will ride for at to finish the path.

The problem gives us the distance, the acceleration due to gravity, and implies that the initial velocity of the picture is zero, as it starts at rest.

We can use the appropriate motion equation to solve for the final velocity:

We can use our values to solve for the time. Keep in mind that the displacement will be negative because the ball is traveling in the downward direction!

The ball will travel upwards to the highest point, then change direction and travel downwards. Remember that the velocity in at the highest point is equal to zero. We can use the following equation and our known data to solve for the initial velocity.

The relationship between constant velocity, distance, and time can best be illustrated as .

We are given the velocity and the distance, allowing us to solve for the time that the ball was in motion.

Isolate the variable for time.

To solve this problem, remember that when something is thrown vertically, its velocity will be  at the highest point. Using this principle, we know the initial velocity, final velocity (zero), and the acceleration. With the appropriate motion equation, we can solve for the distance traveled.

The best equation for this problem is:

Use the given values to solve for the distance.

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.