We fire a missile from ground level at 1750 ft/sec. at launch angle of 10 degrees. h(t) =-16t^2 + vot +h
a. find a function for height as a function
b. find a function for distance from launch point as a function of time.
c. when will the missile reach max height?
d. What is max height?
e. how long will the missile be in the air?
f. how far will it travel?
g. find a function for distance travelled as a function of launch angle
h what is max distance? what launch angle maximizes distance?
find a formula for launch angle as a function of distance
Your initial velocity component up which you are calling Vo is 1750 sin 10
so
Vo = 304 ft/s
( I can hardly believe you are using feet not meters but well here g will be 32 ft/s^2)
Your distance down range, x = u t
where your constant horizontal speed u = 1750 cos 10 (that is part b by the way)
so
u = 1723 ft/s
In your equation if you fire at ground level the Hi on the right is zero. That is the initial height
so
h = -16 t^2 + 304 t
now if we used physics and calculus we would zip ahead, but I suspect you are doing algebra 2 so we will do completing the square on the parabola that will result from plugging in x = u t so t = x/u
h = -16 (x^2/u^2) + 304 (x/u)
since u = 1723
1723^2 h = -16 x^2 +304(1723) x
complete the square on that to find the vertex of the parabola. That is max h and x at max h
Of course t at max h = x/u
Then twice that x and twice that time is when h = zero, ground
that gives you the range and time in the air.
I am going to leave it to you to discover that max range is when you fire at 45 degrees from horizontal.
By the way a physicist would use:
v = Vo - 32 t
and when v = 0, you are at the top
To answer these questions, we will use the given information and apply the laws of projectile motion. Let's break down each question and explain how to find the answers:
a. To find the function for height as a function of time, we use the formula h(t) = -16t^2 + vot + h. In this equation, h(t) represents the height at time t, vo is the initial velocity (1750 ft/sec), and h is the initial height (which is not given). So, the function for height as a function of time is h(t) = -16t^2 + 1750t + h.
b. To find the function for the distance from the launch point as a function of time, we need to consider the horizontal motion. The horizontal distance remains constant over time, as there is no acceleration horizontally. Therefore, the function for distance is simply d(t) = vot.
c. The missile will reach its maximum height when its vertical velocity becomes zero. This occurs at the peak of the trajectory. To find the time when this happens, we need to calculate the time it takes for the vertical velocity to become zero. The vertical velocity at any given time t is given by v(t) = vo - 32t, where 32 ft/sec^2 represents the acceleration due to gravity. Setting v(t) = 0, we can solve for t to find the time when the missile reaches its maximum height.
d. To find the maximum height, we substitute the time found in the previous step (c) into the equation for height as a function of time, h(t). This will give us the maximum height reached by the missile.
e. To determine how long the missile will be in the air, we need to find the time it takes for the object to hit the ground. Since the initial height is not given, we assume the missile lands at ground level (h = 0). We can solve the equation h(t) = 0 to find the time at which the missile lands.
f. The distance traveled by the missile can be found by substituting the time found in step (e) into the equation for distance as a function of time, d(t). This will give us the total horizontal distance traveled by the missile.
g. To find a function for the distance traveled as a function of launch angle, we need to consider the horizontal component of the initial velocity. The horizontal component is given by vo*cos(angle), where angle represents the launch angle. So, the function for distance traveled as a function of launch angle is d(angle) = (vo*cos(angle))*t. Note that time (t) is common to all angles.
h. To determine the maximum distance, also known as the range, we need to find the launch angle that maximizes the function for distance traveled. This can be done analytically or through numerical methods like optimization algorithms.
To find a formula for launch angle as a function of distance, we need to rearrange the equation for distance traveled as a function of launch angle to solve for the angle. This may involve using inverse trigonometric functions, such as arccosine.
Please note that to calculate numerical values for these parameters, you will need to know the values of vo and h.