im trying to help my son any help would be great When a ball is thrown up into the air, it makes the shape of a parabola. The equation S= -16t^2 + v*t + k gives the height of the ball at any time, t in seconds, where “v” is the initial velocity (speed) in ft/sec and “k” is the initial height in feet (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 feet/sec) and any initial height (in feet) 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 feet in the air if it leaves your hand at 5 feet. 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 feet/sec) J
Let's say we have a scenario where a ball is thrown into the air from the ground level, with an initial velocity of 10 ft/sec and an initial height of 2 ft (as if thrown from a person's hand).
Now, let's substitute the values of "v" and "k" in the given equation S = -16t^2 + vt + k.
Given: v = 10 ft/sec, k = 2 ft.
The equation becomes:
S = -16t^2 + 10t + 2.
To find the height of the ball at any two values of time, t, we can substitute different values of t into the equation and calculate the corresponding heights, S.
Let's choose two arbitrary values for time, t:
1. Let's say we want to find the height at 1 second:
Substituting t = 1 into the equation:
S = -16(1)^2 + 10(1) + 2
S = -16 + 10 + 2
S = -4 ft.
2. Now, let's find the height at 3 seconds:
Substituting t = 3 into the equation:
S = -16(3)^2 + 10(3) + 2
S = -16(9) + 30 + 2
S = -144 + 32
S = -112 ft.
So, the height of the ball at 1 second is -4 ft, and at 3 seconds is -112 ft.
In the context of the original problem, the negative heights indicate that the ball has already hit the ground. Since the ball was thrown into the air, it would reach its highest point and start descending, eventually hitting the ground.
In this specific scenario, at 1 second, the ball has reached a height of -4 ft, meaning it is already on its way down after reaching its highest point.
At 3 seconds, the ball has descended further and is now at a height of -112 ft, still on its way down towards the ground.
It's essential to note that these are just illustrative values for the purpose of this explanation, and in real-world scenarios, the initial velocity and height would determine the height and time at which the ball hits the ground.