There are two forces in play in this scenario: friction and gravity. We can use Newton's second law to develop an expression for the acceleration of the block:

Substituting in expressions for each force, we get:

Canceling out mass and rearranging for acceleration, we get:

We know all of our values, allowing us to solve for the acceleration:

We can use the equation for conservation of energy to solve this problem:

If we assume the initial state is when the slingshot is fully loaded and the final state is at the ball's maxmium height, we can completely eliminate kinetic energy to get:

Substituting in our value for initital potential energy and an expression for final potential energy, we get:

Rearranging for final height we get:

The basic kinematics equation for motion under constant acceleration is:

We do not have the acceleration, but we can find it:

Apply this to the first equation and solve.

The equation for speed is:

The runner has gone around the track 1.5 times, thus his total distance traveled is:

We can find the elapsed time from the walker: Rearrange the speed equation to find time:

Now we have all the information we need to find the runner's speed:

If an object is moving to the right, then its displacement is positive. If its displacement is positive then its velocity must be positive as well. The object is speeding up. This implies that the magnitude of its velocity is increasing. If this is true we have that the velocity of the object, , after some time, , has passed must be a larger positive velocity than say, , the object's velocity before that time passed. Therefore, the object's average acceleration is given by:

 since 

So we have that the object's acceleration is positive. If the object's acceleration is positive, then there must be a positive net force causing the acceleration, due to Newton's first law: . Therefore, it is false that the object experiences a net force of 0N.

By definition, acceleration is the rate of change of velocity with respect to time. The average acceleration of an object is given by:

Therefore, if velocity is constant it must be the case that acceleration is 0, since the initial and final velocities of the object are equal. Similarly, if acceleration is non-zero, then it must be the case that the initial and final velocities of the object are different (so velocity is changing). Now, when an object's acceleration and initial-velocity have the same sign (i.e. the same direction) then the object will speed up. In other words, the absolute value of its velocity (the magnitude) increases. This is easy to see if we solve for the final velocity in our equation:

If both  and  have the same sign, then the absolute value of  will be greater as time passes, meaning the object is speeding up. (Try plugging in some numbers). Similarly, when  and  have opposite signs, the magnitude of the object's velocity must start to decrease. After enough time it might even come to a stop, and if it continues to experience the acceleration it will change direction and start gaining velocity on the opposite direction. Regardless, it must first slow down (think about when you throw an object on the air).

Example: 

Example: 

Remember that the positive and negative signs indicate the direction of the vectors. So when it comes to slowing down or speeding up, we want to look at the magnitude of the velocity. Therefore, the only statement that is false is the one that says that if an object has both negative initial velocity and acceleration it will slow down. The object will speed up and continue to move to the left. 

If the object is moving to the left, then its displacement must be negative since its final position will be further to the left in than its initial position. So in an imaginary number line, after some time passes, the object's final position is less than the object's initial position. This can be shown mathematically:

 , since

Since the object's displacement is negative, then so is its velocity. This can easily be seen from the equation for average velocity:

 since . At the end of its trajectory, the object stops, so overall the object's velocity is less than or equal to .

The object eventually stops. This means that the magnitude of the object's velocity is decreasing. In other words, the absolute value of the object's velocity is becoming smaller and smaller until it stops. So, for example, if at some moment the object's velocity is , after some time its velocity will be . Remember that the negative sign indicates only the direction of the vector (the object is moving to the left, hence its velocity is negative). But in absolute terms, the object is moving slower and slower (to the left) until it stops. Finally, we can see that the acceleration must be positive in order to slow down an object moving to the left. Going of our example, let's say the object is moving at and after some time it slows down to  (again the negative sign just indicates that the object moves to the left). So, if look at the average acceleration we have:

Thus the acceleration must be positive.

A trick that I often use is if something is speeding up, the acceleration and velocity point in the same direction. If something is slowing down. the acceleration and velocity point opposite directions. Since the elevator is speeding up, the acceleration must point in the same direction as it's velocity: downwards.

The origin is placed at the location the ball is thrown and caught so y=0 when it is thrown and caught. Because the ball only moves in one dimension (up or down), the velocity must drop to zero when it transitions from a upward positive velocity to a downward negative velocity at the apex. The acceleration on the surface of earth is always g for projectiles when neglecting air resistance or other forces.

The slope of a distance vs. time plot represents the object's velocity. Therefore, in both the first and second graphs, the velocity is constant and zero, respectively. An object with constant (or no) velocity is experiencing, by definition, no net force. Oppositely, in the third graph the slope is increasing, meaning the object is accelerating, meaning it must be subject to some force.