# Time vs Height - EASY but easily forgotten :/

• Synix
In summary, Eugenia accidentally dropped her cell phone into a construction site and asked the construction manager, Tony, to throw it back up to her. The height of an object thrown vertically up is modeled by the equation s(t) = s0 + v0t - 1/2gt^2, where s0 is the initial height, v0 is the initial velocity, g is the acceleration due to gravity, t is the time, and s(t) is the height at time t. To determine the initial velocity needed for the cell phone to reach Eugenia, the equation v_f^2 = v_i^2 + 2aΔs can be used. At the top of its flight, the velocity is zero
Synix
1.Homework Statement
One day, while working on a downtown construction site, Eugenia accidentally dropped her cell phone into the large excavation on the site. Fortunately, the cell phone landed on a pile of soft Earth and was not damaged. She asked Tony, the construction manager, who was working nearby to throw the cell phone back up to her.

The height of an object thrown vertically up is modeled by

s(t) = s0 + v0t - 1/2gt^2

where s0 is the initial height above the ground, v0 is the initial velocity, g is the acceleration due to gravity, t is the time, and s(t) is the height above the ground at time t.

If Tony is 10m below ground level, and the acceleration due to gravity is 9.8m/s^2, with what initial velocity must he throw the cell phone for it to reach Eugenia? Explain any assumptions you make and show your calculations, graphs, or both to justify your answer.

## Homework Equations

h(t) = t^2 + t + c

s(t) = s0 + v0t - 1/2gt^2

## The Attempt at a Solution

Fail.
http://i72.photobucket.com/albums/i169/Alkapwn/math2-1.jpg

Last edited:
Hello Synix, welcome to PF!

Are you familiar with the equation $$v_f^2 = v_i^2 + 2a\Delta s$$ ?

Thats the equation we need for this one. After Tony throws up the cellphone, when it reaches Eugenia it should be at the top of its flight. At this point, was is its velocity? I'm sure you can figure it out from here.

I can understand how easily it is to forget certain equations or concepts, especially when they are not used frequently. However, it is important to always double check and refresh our memory before attempting to solve a problem. In this case, the equation s(t) = s0 + v0t - 1/2gt^2 is the correct one to use, as it models the height of an object thrown vertically up.

In order to solve this problem, we need to make some assumptions. First, we will assume that Tony throws the cell phone with an initial velocity (v0) directly upwards, as this is the most efficient way to get the cell phone back to Eugenia. Second, we will assume that the initial height (s0) of the cell phone is 10m, since Tony is 10m below ground level. Finally, we will use the known value for the acceleration due to gravity, which is 9.8m/s^2.

Now, we can plug in our values into the equation s(t) = s0 + v0t - 1/2gt^2 and solve for v0.

s(t) = 10m + v0t - 1/2(9.8m/s^2)t^2

Since we want the cell phone to reach Eugenia, we can set the final height (s(t)) equal to 0.

0 = 10m + v0t - 1/2(9.8m/s^2)t^2

Solving for v0, we get:

v0 = 4.9t^2/t - 10m/t

Since t is the time it takes for the cell phone to reach Eugenia, we can estimate this value to be around 1 second. Therefore, v0 = 4.9m/s - 10m/s = -5.1m/s.

In conclusion, Tony must throw the cell phone with an initial velocity of -5.1m/s in order for it to reach Eugenia. It is important to always double check our assumptions and equations to ensure accurate and precise solutions.

## 1. What is the relationship between time and height?

The relationship between time and height is that as time passes, an object's height will change. This change in height can be either an increase or decrease, depending on the direction of motion.

## 2. How does gravity affect the time-height relationship?

Gravity plays a key role in the time-height relationship. It is the force that causes objects to fall towards the Earth, thus affecting their height as time passes. This is known as acceleration due to gravity.

## 3. Does an object's mass affect its time-height relationship?

Yes, an object's mass can affect its time-height relationship. Objects with a larger mass require more force to move, so they may take longer to reach a certain height compared to objects with a smaller mass.

## 4. Is there a limit to how high an object can go in a given amount of time?

Yes, there is a limit to how high an object can go in a given amount of time. This is due to the effects of air resistance and gravity, which can slow down an object's ascent and prevent it from reaching an infinite height.

## 5. How does air resistance affect the time-height relationship?

Air resistance, also known as drag, can affect the time-height relationship by slowing down an object's motion. This means that it may take longer for an object to reach a certain height due to the resistance it experiences from the air.

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