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307-E: Pop-Up 1: Numeric Analysis

• Topic Cluster: Kinematics
• Topic: Projectile Motion
• Objective: Solve numeric problems about objects falling down and objects thrown directly upwards.
• Content: Objects thrown upwards have an acceleration of 9/8 m/s^2 downwards, have an initial velocity that is upwards, and follow the four kinematic equations
• Level: 2

In this section, we will epxlore pop-up problems. Pop-up problems are a good step. They build on previous kinematics problems, in particular, falling problems, and they lead into more complicated projectile motion problems

What is free-fall?

A an object is in free-fall when the only force acting on that object is gravity, and the object is accelerating downward at 9.8 m/s2. An object does not need to be falling to be in free-fall! It can be rising up or even moving horizontally, as long as it is in the air and not touching anything.

In a pop-up problem, the acceleration is consant (9.8 m/s2 downward), and thus, all of the kinematic equations apply:

Also consider these concepts

• In all of those equations, the variable $$\Delta y$$ is used instead of $$\Delta x$$, because the object moves in the vertical, not horizontal, dimension.
• The object's acceleration is -9.8 m/s2. This value is negative because it is downward.
• When an object is on the way up, its velocity is positive.
• When the object reaches the very top, it momentarily has velocity zero.
• When an object is on the way down, its velocity is negative
1. When the object is on the way up, which of the following is true?
1. Velocity and acceleration are in the same direction, and the object speeds up.
2. Velocity and acceleration are in opposite directions, and the object slows down.
3. Acceleration is zero and the object moves at a constant velocity.
2. When the object is on the way down, which of the following is true?
1. Velocity and acceleration are in the same direction, and the object speeds up.
2. Velocity and acceleration are in opposite directions, and the object slows down.
3. Acceleration is zero and the object moves at a constant velocity.

Upward motion only

Many pop-up problems deal only with flying upward, and ignore the downward motion. Two questions like this are: "How high does an object rise?" and "How long does it take the object to reach its highest point?" The following concepts can

• If a question asks "How high does an object rise?", you are looking for vertical displacement.
• If a question asks "How long does it take to reach its highest point?", you are looking for change in time.
• At the very top of the motion, the object momentarily has velocity zero. So if you are analyzing motion from the moment it is thrown upward to the top, the final velocity is equal to zero!
1. An object is thrown directly upward with a speed of 25 m/s. How much time does it take to reach the highest point?
2. An object is thrown directly upward with a speed of 5 m/s. How much time does it take to reach the highest point?
3. An object is thrown directly upward with a speed of 16 m/s. How high does it rise into the air?
4. An object is thrown directly upward with a speed of 30 m/s. How high does it rise into the air?

Downward and upward motion landing in the same spot

Several other problems deal with an object that is thrown directly upward, rises very high up, then falls down in the same place that it started, on the same exact height.

When an object is thrown directly upward, it spends exactly as much time going up as it spends going down. Thus, the total time in the air is twice as long as the total time it took the object to rise. Therefore, you can use the same method as in the previous problems to answer these problems.

1. An object is thrown directly up into the air with a speed of 49 m/s. It rises and falls and then lands in the same spot.
1. Determine the time necessary for the object to reach its highest point.
2. The total time in the air is twice the time to reach the highest point. Determine this total time.
3. Determine the velocity of the object right before it strikes the ground.
4. Determine the total displacement of the object in its motion.
2. An object is thrown directly up into the air with a speed of 22 m/s. It rises and falls and then lands in the same spot.
1. Determine the time necessary for the object to reach its highest point.
2. The total time in the air is twice the time to reach the highest point. Determine this total time.
3. Determine the velocity of the object right before it strikes the ground.
4. Determine the total displacement of the object in its motion.

When you did those problems above, you proved some very important principles about pop-up motion!

• Pop-up motion is always symmetic. This means that the total time that the object is in the air is twice the time that it takes to go all the way up.
• Pop-up motion is always symmetic. This means that the final velocity, when the object strikes the ground, is precisely the negative of the initial velocity.
• Because, in these problems, an object rises up and then falls to eactly the same spot, the vertical displacement $$\Delta y$$ is equal to zero.

In the next set of problems, you will use these principles to figure out the total time with three different methods!

1. An object is thrown directly into the air at a speed of 18 m/s, rises up, and then lands. How long does it spend in the air? Please answer this question in three different ways, and show that you get the same answer each time:
1. Find the total time it takes to reach the top of its motion, and double that time.
2. Find the time it takes if the final velocity is precisely the negative of the initial $$v_f = -18 \text{ m/s}$$.
3. Find the time if the final displacement is equal to zero.
2. An object is thrown directly into the air at a speed of 44 m/s, rises up, and then lands. How long does it spend in the air? Please answer this question in three different ways, and show that you get the same answer each time:
1. Find the total time it takes to reach the top of its motion, and double that time.
2. Find the time it takes if the final velocity is precisely the negative of the initial $$v_f = -18 \text{ m/s}$$.
3. Find the time if the final displacement is equal to zero.
3. If you use the information $$\Delta y = 0$$, you get two solutions to your mathematical equation. One of them is the total time in the air, and the other is $$\Delta t = 0$$. Why?