A water rocket was launched from the ground, with an initial velocity of 32m/s. The rocket achieved a height of 44 m after 2 s of flight. The rocket was in the air for 6 s. Determine the quadratic functionthat models the height of the rocket over time.
you sure that h(2) = 44?
I suspect it is h(3) = 44.
To determine the quadratic function that models the height of the rocket over time, we can use the general form of a quadratic function:
y = ax^2 + bx + c
where y represents the height of the rocket and x represents the time.
We are given the following information:
Initial velocity (u) = 32 m/s
Height after 2 seconds (h1) = 44 m
Total time in the air (t) = 6 seconds
To determine the quadratic function, we need to find the values of a, b, and c.
Step 1: Determine the value of a:
a is the coefficient of x^2 in the quadratic function. Since gravity acts in the vertical direction, which is modeled by a quadratic function, a is related to gravity and is equal to half the acceleration due to gravity (g/2). In this case, g = 9.8 m/s^2.
a = g/2 = 9.8/2 = 4.9
Step 2: Determine the value of b:
b is the coefficient of x in the quadratic function. It represents the initial velocity (u) of the rocket. In this case, u = 32 m/s.
b = u = 32
Step 3: Determine the value of c:
c is the constant term in the quadratic function. It represents the initial height (h0) of the rocket when it was launched from the ground. To find c, we can substitute the known values of a, b, and h1 into the quadratic function and solve for c.
h1 = a(2)^2 + b(2) + c
44 = 4.9(4) + 32(2) + c
44 = 19.6 + 64 + c
c = 44 - 19.6 - 64
c = -39.6
Step 4: Write the quadratic function:
Now that we have the values of a, b, and c, we can write the quadratic function that models the height of the rocket over time.
y = ax^2 + bx + c
y = 4.9x^2 + 32x - 39.6
Therefore, the quadratic function that models the height of the rocket over time is y = 4.9x^2 + 32x - 39.6.
To determine the quadratic function that models the height of the rocket over time, we need to understand the general equation for the height of a rocket in projectile motion.
The equation for the height of an object in projectile motion can be represented as:
h(t) = -gt^2 + v₀t + h₀
Where:
- h(t) is the height of the object at time t.
- g is the acceleration due to gravity (approximately -9.8 m/s²).
- v₀ is the initial velocity of the object.
- h₀ is the initial height of the object.
Given that the rocket achieved a height of 44 m after 2 seconds and was in the air for a total of 6 seconds, we can use this information to find the values of v₀ and h₀.
Using the equation h(t) = -gt^2 + v₀t + h₀, we can substitute the values we know:
h(2) = -9.8(2)^2 + v₀(2) + h₀ = 44
h(8) = -9.8(8)^2 + v₀(8) + h₀ = 0
We now have two equations with two unknowns (v₀ and h₀). Let's solve them simultaneously.
From the first equation:
-9.8(4) + 2v₀ + h₀ = 44
From the second equation:
-9.8(64) + 8v₀ + h₀ = 0
We can subtract the first equation from the second equation to eliminate h₀:
-9.8(64 - 4) + 8v₀ - 2v₀ = 0 - 44
Simplifying:
-9.8(60) + 6v₀ = -44
Divide both sides by 6:
v₀ = (44 + (9.8 * 60))/6
Now, substitute this value of v₀ back into the first equation to find h₀:
-9.8(4) + 2v₀ + h₀ = 44
Simplifying:
h₀ = 44 + 19.6 - 2v₀
Substituting the value of v₀ into h₀:
h₀ = 44 + 19.6 - 2 * ((44 + (9.8 * 60))/6)
Simplifying further, we get the value of h₀.
Once we have the values of v₀ and h₀, we can write the quadratic function that models the height of the rocket over time as:
h(t) = -4.9t^2 + v₀t + h₀
Note: The equation h(t) = -4.9t^2 instead of -9.8t^2 is used because the equation assumes that the upward direction is considered positive, so the acceleration due to gravity (g) is divided by 2.