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.

This is a velocity versus time graph.  The displacement can be determined from the area under the curve of the graph.

Begin with the first .  This is a triangle with a base of  and height of .

Next look at the next .  This is a triangle with a base of  and height of .  That negative is important as it indicates that the object was traveling in the opposite direction.

For the final , the area is that of a rectangle with a base of  and a height of . Again this negative is important as it indicates what direction the object was traveling in.

To find the total displacement, add all of the areas together.

Therefore the final displacement is .

This is a position versus time graph.  The total distance traveled is the total position that has been traveled during the entire time.

From 0 to 10 seconds, the object traveled .

From 10 seconds to 15 seconds, the object did not move.

From 15 seconds to 30 seconds the object traveled  backward.

The distance is the total amount traveled.

The displacement which is the distance traveled from the original position.  Since at  the object has returned to the , the displacement is .

This is a position versus time graph.  To find the velocity of an object at a particular time, determine the slope of the graph at that time.  Using the points  and  we can determine the slope of the graph at 5 seconds since it is a constant slope.

The important thing to note is that the question asks for the velocity at any given point. The average velocity will be equal to the total displacement divided by the total time, but the question is asking for the instantaneous velocity.

Velocity is calculated by a change in displacement over a change in time. In a displacement versus time graph, this is equal to the slope.

This tells us that the slope at a certain time will be equal to the velocity at that time.

In calculus terms, velocity is the derivative of the function for displacement in terms of time. What this means is that at any point on the graph, the instantaneous slope is the velocity for that given time. To determine velocity, one must find the slope of the line at that particular time interval.

The important thing to note is that the question asks for the acceleration at any given point. The average acceleration will be equal to the net change in velocity divided by the total time, but the question is asking for the instantaneous acceleration.

Acceleration is calculated by a change in velocity over a change in time. In a velocity versus time graph, this is equal to the slope.

This tells us that the slope at a certain time will be equal to the acceleration at that time.

In calculus terms, acceleration is the derivative of the function for velocity in terms of time. What this means is that at any point on the graph, the instantaneous slope is the acceleration for that given time. To determine acceleration, one must find the slope of the line at that particular time interval.

This is a velocity versus time graph.  During the first 5 seconds, the velocity is approaching 0.  This means that the object is slowing down.

During the next 5 seconds, the velocity is increasing in the negative direction.  This means that the object is speeding up in the negative direction.

During the final 5 seconds, the velocity value does not change.  This means that the object is remaining at a constant velocity.