The height (in feet) of a projectile shot vertically upward from a point 6 ft above ground level

The height (in feet) of a projectile shot vertically upward...

Key Concepts

Finding Maximum Height via Critical Points

The maximum height in projectile motion is reached at the critical point where the velocity becomes zero. Setting the derivative of the displacement function equal to zero allows one to solve for the time at which the projectile reaches its peak, after which the height can be determined by evaluating the displacement function at that time.

Solving Quadratic Equations

Quadratic equations arise naturally in projectile motion problems, particularly when setting the position function equal to a specific value (such as ground level) or when determining the time at which the projectile reaches a particular height. Solving these equations helps to find critical times in the object's trajectory.

Sign Analysis of Velocity

The sign of the velocity function indicates the direction of motion. A positive velocity means that the object is moving upward, while a negative velocity indicates downward motion. This analysis is key in determining the time intervals during which different phases of the motion occur.

Differentiation in Kinematics

Differentiation is used to obtain the velocity function from the displacement function. By taking the derivative of the position function with respect to time, one can determine the instantaneous velocity of the projectile, which is crucial for analyzing when the projectile is rising or falling.

Kinematics Equations

Kinematics equations relate an object's displacement, velocity, acceleration, and time. In problems involving projectile motion, these equations enable the determination of key parameters like maximum height, time of flight, and the conditions under which the object ascends or descends.

Projectile Motion

Projectile motion involves the movement of an object under the influence of gravity, typically following a parabolic path. This concept uses kinematics equations to describe the position, velocity, and acceleration of the object in motion, making it fundamental for analyzing objects launched into the air.

Transcript

00:01 Given the position vector minus, well, we're told basically that we're actually given a scenario here, right? we'll have a projectile shot vertically upward from six feet off the ground.

00:13 The position, which is, you know, is upward, is given by minus 16 t squared plus 48t plus 6.

00:25 The velocity then is minus 32t plus 48, so at t equals zero, we shot it up at, that velocity of 48, did they give us any units feet? i guess feet per second, 48 feet per second? yeah, they gave us feet, but they didn't really explicitly tell us that they were seconds.

00:46 But yeah, it's going to be seconds because this is 32 .2g is about 32 feet per second squared.

00:56 So if this were, you know, an accurate scenario on earth, then, you know, that would be the gravitation.

01:02 Acceleration.

01:05 So let's see here.

01:06 We they want to know, let's see, we need to figure out when its velocity is less than zero and greater than zero, so we can figure out when the velocity equals a zero.

01:15 And that we just solve for that and we get t is three halves.

01:20 So t1, we called it.

01:22 So at three halves of a second, at 1 .5 seconds, it's zero.

01:29 And so if we look, it's greater than zero between zero and three halves, 1 .5.

01:38 And then, where did i get? why did i write two there? three halves.

01:46 It's less than zero as we go from three half, one and a half to whenever it hit the ground again.

01:54 They call it capital t.

01:57 We could figure that out, but they didn't ask us to figure that out.

02:01 Obviously, we set this equal to zero and figure that out.

02:06 Now, we can see, then they ask us for the maximum height, and that's just the maximum height occurs when the velocity is zero.

02:17 And so we can plug this back into here, and we get that that is 42 feet.

02:22 So we go up 42 feet.