Physics tells us that an object projected at time t=0
 from a point y0
 feet above ground with an initial vertical velocity of v0
 feet per second has a height above the ground after t seconds that is modeled by the function h(t)=−16t2+v0t+y0
. At that same time t, the distance the object has traveled is modeled by the function d(t)=v1t
 where v1
 is the initial horizontal velocity. 

Suppose a projectile is fired from a launcher 5
 feet off of the ground with an initial vertical velocity of 100ft/s
 and an initial horizontal velocity of 115ft/s
. Find the height of the projectile and its horizontal distance from the launcher after 3
 seconds.


The problem provides two key functions:

1. The height of the projectile at time \( t \), given by:
   \[
   h(t) = -16t^2 + v_0 t + y_0
   \]
   where:
   - \( v_0 = 100 \, \text{ft/s} \) (initial vertical velocity),
   - \( y_0 = 5 \, \text{ft} \) (initial height from the ground).

2. The horizontal distance of the projectile at time \( t \), given by:
   \[
   d(t) = v_1 t
   \]
   where:
   - \( v_1 = 115 \, \text{ft/s} \) (initial horizontal velocity).

### To find:
1. **Height of the projectile after 3 seconds:**
   \[
   h(3) = -16(3)^2 + 100(3) + 5
   \]

2. **Horizontal distance from the launcher after 3 seconds:**
   \[
   d(3) = 115(3)
   \]

Let me calculate both values.After 3 seconds:

1. The height of the projectile is \( 161 \, \text{feet} \).
2. The horizontal distance from the launcher is \( 345 \, \text{feet} \).

Would you like further details or explanations on any part of this?

Here are 5 related questions you might explore:
1. What would happen to the height if the initial vertical velocity was different?
2. How long would the projectile remain in the air before hitting the ground?
3. How far horizontally would the projectile travel before landing?
4. How does the acceleration due to gravity (-16 ft/s²) affect the height over time?
5. What if the projectile was launched from a higher platform, how would that impact the motion?

**Tip:** When dealing with projectile motion, splitting the problem into vertical and horizontal components helps simplify the analysis.

Suppose a projectile is fired from a launcher 5 feet off of the ground with an initial vertical velocity of 100 ft/s and an initial horizontal velocity of 115 ft/s. Find the height of the projectile and its horizontal distance from the launcher after 3 seconds.

Explore how to calculate the height and horizontal distance of a projectile launched from 5 feet above the ground with an initial vertical velocity of 100 ft/s and horizontal velocity of 115 ft/s after 3 seconds. Learn about projectile motion, quadratic functions, and kinematic equations in this detailed solution. Suitable for students in Grades 9-12.

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