When a ball is thrown up into the air, it makes the shape of a parabola. The equation S= -10t^2 + v*t + k gives the height of the ball at any time, t in seconds, where “v” is the initial velocity (speed) in meters/sec and “k” is the initial height in meters (as if you were on top of a tower or building).
Make up a scenario where a ball is thrown, shot, etc. into the air. You can choose any initial velocity (in meters/sec) and any initial height (in meters) of the ball, but include them in your written scenario. The ball can leave your hand, the top of a building, etc. so you can use many different values for the initial height.
Insert the chosen values for “v” and “k” into the formula listed above.
Use the formula to find the height of the ball at any two values of time, t, in seconds that you want. Show your calculations and put units on your final answer!
Provide a written summary of your results explaining them in the context of the original problem.
Please make sure that your answers make sense!
If your answer is negative, that means the ball already hit the ground, so choose a smaller value for time.
Think about a ball going up into the air, you might throw it or put in a cannon. If you throw a ball up into the air, it will not end up being 800 meters in the air if it leaves your hand at 5 meters. Therefore, you would need to adjust your initial velocity. You may want to research initial velocity (speed) to figure out what seems reasonable! (ex. Your 5 year old cannot throw a ball into the air with an initial velocity of 300 meters/sec) !
You could plagiarize Longfellow ,
"I shot an arrow into the air,
It fell to earth, I knew not where;
For, so swiftly it flew, the sight
Could not follow it in its flight.
I breathed a song into the air,
It fell to earth, I knew not where;
For who has sight so keen and strong,
That it can follow the flight of song?
Long, long afterward, in an oak
I found the arrow, still unbroke;
And the song, from beginning to end,
I found again in the heart of a friend."
with ...
I kicked a ball into the air,
It fell to earth, I knew not where,
but I know it's initial velocity was 24 m/sec and it's initial height was .1 m.
Now follow the instructions given to you.
Scenario: A ball is thrown upward from the top of a building with an initial velocity of 20 m/s. The ball is thrown from a height of 30 meters above the ground.
Given:
Initial velocity (v) = 20 m/s
Initial height (k) = 30 meters
Let's calculate the height of the ball at two different time values.
1. For t = 2 seconds:
Substitute t = 2 into the equation S = -10t^2 + vt + k.
S = -10(2)^2 + 20(2) + 30
S = -40 + 40 + 30
S = 30 meters
2. For t = 4 seconds:
Substitute t = 4 into the equation S = -10t^2 + vt + k.
S = -10(4)^2 + 20(4) + 30
S = -160 + 80 + 30
S = -50 meters
Summary:
At t = 2 seconds, the ball is at a height of 30 meters above the ground, which is the initial height of the ball.
At t = 4 seconds, the ball is at a height of -50 meters, which means the ball has already hit the ground.
In this scenario, the ball reaches its maximum height and starts descending after around 2 seconds. Therefore, it makes sense for the ball to hit the ground at t = 4 seconds. The negative value of -50 meters indicates that the ball is below the initial height and has indeed hit the ground.
Scenario:
Let's say we have a ball that is shot straight up into the air from the ground with an initial velocity of 30 meters/second. The ball is shot from the rooftop of a building that is 50 meters high.
Given:
Initial velocity (v) = 30 meters/second
Initial height (k) = 50 meters
Equation: S = -10t^2 + vt + k
To find the height of the ball at two different values of time, let's choose t = 2 seconds and t = 4 seconds.
Calculations:
1. For t = 2 seconds:
S = -10(2^2) + 30(2) + 50
S = -40 + 60 + 50
S = 70 meters
The height of the ball at t = 2 seconds is 70 meters.
2. For t = 4 seconds:
S = -10(4^2) + 30(4) + 50
S = -160 + 120 + 50
S = 10 meters
The height of the ball at t = 4 seconds is 10 meters.
Summary:
In this scenario, when the ball is shot straight up into the air from the rooftop of a building with an initial velocity of 30 meters/second and an initial height of 50 meters, we found that at t = 2 seconds, the ball is at a height of 70 meters. At t = 4 seconds, the ball has reached a height of 10 meters.
Since the parabolic equation describes the height of the ball at any given time during its flight, we can use it to calculate the height at different time intervals. It is important to note that the height will eventually decrease and become negative, indicating that the ball has hit the ground.