Algebra Word Problem Projectile Motion Problem
You know that a dropped object falls a distance of 16t2feet in t seconds. When an object is not simply released but is thrown or launched, it is called a projectile. What happens to a projectile that is launched with some initial vertical velocity, Vo (measured in feet per second), at some initial height ho(measured in feet)?
Without gravity to pull the projectile its height hWould increase according to the equation h = vot + ho • With gravity, the projectile falls 16t2 feet in t seconds. • So the projectile’s height at any time t is given by H = -16t2 + v0t + h0
In order to solve quadratic equations involving maximums and minimums for projectile motion, it is necessary to • know how to solve quadratic equations: • know vertex formula for a parabola • write and solve an equation for the problem
Let’s solve the example of a quadratic equation involving maximums and minimums for projectile motion • An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in meters. When does the object strike the ground?
What is the height (above ground level) when the object smacks into the ground? • Well, zero, obviously. • So I'm looking for the time when the height is s = 0. • I'll set s equal to zero, and solve: • 0 = –4.9t2 + 19.6t + 58.8
0 = –4.9t2 + 19.6t + 58.8 0 = t2 – 4t – 12 0 = (t – 6)(t + 2) Then t = 6 or t = –2.
The second solution is from two seconds before launch, which doesn't make sense in this context. • So "t = –2" is an extraneous solution, and I'll ignore it.
Answer: The object strikes the ground six seconds after launch.
Physics Exploration You’re playing Angry Birds on your sweet new iPhone and get curious about the maximum height, distance, and time the bird is in the air before it hits the ground again. This formula is from Science class :
Key Information Box: t = Time (in seconds) Our unknown variable in most cases g = Gravity 4.9 if unit is meters, 16 if unit is feet V0 = Initial velocity either m/s or ft/s h0 = InitialHeight of objectfeet or meters
If you launch a bird at an initial velocity of 10 m/s from the ground, how long will it take until the bird hits the ground again? • What do you know? • v0= 10 m/s • g = 4.9 (because we’re in meters) • h0= 0 because FROM GROUND
If you launch a bird at an initial velocity of 10 m/s from the ground, how long will it take until the bird hits the ground again? • What do you WANT to know? • time until s(t) = 0 again.
A = -4.9 • B = 10 • C = 0 • Answer: Hits ground again 2.04 sec. after launch. • Key Question: Why not the negative?
Another Problem: • An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. What will be the object's maximum height? When will it attain this height?
Hmm... They didn't give me the equation this time. But that's okay, because I can create the equation from the information that they did give me.
The initial height is 80 feet above ground and the initial speed is 64 ft/s. Since my units are "feet", then the number for gravity will be 16, and my equation is: • s(t) = –16t2 + 64t + 80
They want me to find the maximum height. For a negative quadratic like this, the maximum will be at the vertex of the upside-down parabola. So they really want me to find the vertex.
t = –b/2a = –(64)/2(–16) = –64/–32 = 2 s = s(2) = –16(2)2 + 64(2) + 80 = –16(4) + 128 + 80 = 208 – 64 = 144 • And in this case, the vertex is at (2, 144).
But what does this vertex tell me? According to my equation, I'm plugging in time values and extracting height values, so the input "2" must be the time and the output "144" must be the height. • It takes two seconds to reach the maximum height of 144 feet
New Question If you launch a bird at an initial velocity of 10 m/s from the ground, what is the maximum height the bird will reach? When will the bird reach this height? What do we want to know? • VERTEX!
Graph it and find the vertex. • Max height is y-coordinate, time is x-coordinate. • In a sentence: “Bird reaches maximum height of 5.16 meters after 1.06 seconds.
Now… Practice! An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. • Question 1: What is the initial velocity (v0)? • Question 2: What is the initial height (h0)? • Question 3: What is the value for g? • Question 4: What’s our equation? • Question 5: What are the values for a, b, and c? • Question 6: How long will it take before the object hits the ground? • Question 7: What is the maximum height, and after how many seconds will the object reach that height?