An explosion causes debris to rise vertically with an...

An explosion causes debris to rise vertically with an initial velocity of 72 feet per second. The function $$ s(t)=-16 t^{2}+72 t $$ describes the height of the debris above the ground, $s(t),$ in feet, $t$ seconds after the explosion. a. What is the instantaneous velocity of the debris $\frac{1}{2}$ second after the explosion? 4 seconds after the explosion? b. What is the instantaneous velocity of the debris when it hits the ground?

An explosion causes debris to rise vertically with an initial velocity of 72 feet per second. The function
$$
s(t)=-16 t^{2}+72 t
$$
describes the height of the debris above the ground, $s(t),$ in feet, $t$ seconds after the explosion.
a. What is the instantaneous velocity of the debris $\frac{1}{2}$ second after the explosion? 4 seconds after the explosion?
b. What is the instantaneous velocity of the debris when it hits the ground?

Key Concepts

Quadratic Equations

Quadratic equations, which are second-degree polynomial equations, are used to model various phenomena in physics including the parabolic trajectories of projectiles. Solving a quadratic equation can help determine key points in the motion, such as the time when an object reaches the ground.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value is changing. In the context of motion, differentiating the position function with respect to time yields the velocity function, which is essential for determining instantaneous speeds at specific moments.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a particular point in time. It is found by evaluating the derivative of the position function at the given time. This concept is central in kinematics and calculus when analyzing dynamic motion.

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air and influenced primarily by gravity. The trajectory of such an object is typically modeled using quadratic functions, and the analysis of its flight involves understanding concepts like initial velocity, acceleration due to gravity, and the moment when it returns to the ground.

Transcript

00:01 A displacement function s of t that is minus 16 t squared plus 72t.

00:07 And it asks us to find the velocity at half a second.

00:11 So it wants the velocity of one half.

00:13 And it wants the velocity at four seconds.

00:17 Okay.

00:18 So what i have to do is i have to define the velocity.

00:20 And the velocity function is the derivative of the displacement function.

00:26 And i would encourage you to look back at the previous video, the previous questions video, if you want to look at where i'm going to explain the power rule, but essentially this derivative becomes this.

00:38 So the power rule says bring the power down.

00:41 So two times minus 16 is minus 32 and subtract one from the power.

00:44 So t square became t and then plus 72 because the derivative of t is one.

00:49 So all i have to do is plug in a half and then plug in four.

00:56 So half of 32 is minus 16.

01:00 So i get minus 16 plus 72.

01:02 Okay, and then you just have to do 72 minus 16, and 72 minus 16 is 56.

01:08 So we end up with 56 feet per second.

01:11 Then when we do it with four, we plug in minus 32 times four plus 72.

01:19 So that's minus 128 plus 72.

01:23 Okay...