Problems Plus 1: Projectile fired from the origin

Problems Plus 1: Projectile fired from the origin | Ecency

In this video, I examine projectile motion launched from the origin and show that the projectile will always lie within a parabola defined by launching a projectile from the maximum height horizontally at the same initial velocity. I also show that if we are to aim directly at a target downrange and drop the target at the same instance of firing the projectile, then we will always hit our target!

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1 Problem 1 Projectile Origin.png

#math #calculus #vectorfunctions #problemsplus #projectilemotion

Timestamps

Problem 1: Projectile fired from the origin – 0:00
Solution to (a): Angle to achieve maximum height – 2:53
- Maximum height is when the derivative of the position vector = 0 – 3:45
- Time at maximum height – 5:29
- Maximum height is at 90 degrees – 10:00
Solution to (b): Projectile must stay within this parabola – 10:36
- Recap on the time until a projectile hits the ground – 17:15
- |x| is less than or equal to |R| – 21:21
- Subtracting the projectile path by the formula for the parabola – 30:45
- Difference between the projectile path and parabola is always negative or zero – 37:37
- Obtain a quadratic formula involving the point (a, b) on the projectile path inside or on parabola – 50:35
- Discriminant has to be greater than or equal to zero: 54:15
- The height b satisfies the condition of being on or inside the parabola – 58:08
- Point (a, b) lies on the path of the projectile when tan α satisfies the quadratic formula – 1:03:05
Solution to (c): Projectile always hits dropped target – 1:04:41
- Launch angle is aimed at target that gets dropped – 1:08:48
- Target falls due to gravity at the same rate as the projectile – 1:12:36
- Projectile always hits the target! – 1:13:22