(a) In Problem 8 , using energy techniques rather than the techniques of Chapter 4, find the spe

step 1 Hello, all right. So let's do problem number 12. And it's literally, it's just another version o

(a) In Problem 8 , using energy techniques rather than the...

Key Concepts

Mass Independence in Free Fall

In energy conservation calculations dealing with gravitational fields, the mass of the object often cancels out when equating gravitational potential energy and kinetic energy. This indicates that, in the absence of resistive forces, the final speed of a falling or projected object is independent of its mass, which is a key insight in solving such problems.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion, expressed as one half of the mass times the square of the speed (½mv²). When an object falls or is projected, the loss in gravitational potential energy is converted into kinetic energy, which is why energy conservation methods can be used to determine its speed at various points in its trajectory.

Projectile Motion and Launch Angle

Projectile motion involves analyzing the movement of an object launched into the air under the influence of gravity. The launch angle determines the initial distribution of kinetic energy between the horizontal and vertical components of motion; however, when using energy conservation, the total kinetic energy and thus the speed remains the same regardless of the angle if only gravity does work.

Conservation of Mechanical Energy

This concept states that in the absence of non-conservative forces (like friction or air resistance), the total mechanical energy of a system remains constant. In problems involving gravitational motion, the initial kinetic and potential energies transform into final kinetic energy and potential energy without loss, thereby allowing us to compute speeds by equating the initial and final energy states.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically calculated as the product of mass, gravitational acceleration, and height (mgh). In the context of objects moving vertically, changes in gravitational potential energy convert into kinetic energy as the object moves to or from different heights.

Transcript

00:01 Hello, all right.

00:02 So let's do problem number 12.

00:05 And it's literally, it's just another version of problem number eight.

00:10 So remember problem number eight is the snowball problem.

00:13 And that is a snowball is being basically thrown off a cliff right at a angle of 41 degrees, and at a velocity of 40 degrees.

00:30 13 meters per second.

00:34 We know that the height of this cliff is 12 .5 meters.

00:40 Now, this problem wants to know what is the speed when the snowball reaches the ground? that's question a.

00:49 So, you know, like once again, this is a conservation of energy problem.

00:54 So what does that state? conservation of energy states that some of the initial energies is equal to some ooh, that is ugly equal to some of final energies.

01:06 So let's look at what we have for our initial energies, right? we have, obviously, we have gravitational potential energy.

01:13 We also have a initial kinetic energy because this ball is being thrown, i mean, being launched off.

01:23 And then we do not have a potential, we do not have a gravitational potential energy.

01:29 So let's cross that out because it will be reaching ground level and at ground level height is zero but we do have a final kinetic energy and now let's let's expand this so gravitational potential energy we have mgh plus one half m and it's v initial squared right but what is v initial square? well, it is actually the cosine of 41 so the initial squared is cosine of 41 degrees times 14 squared times 14 meters per second squared is equal to one half mv final squared simple all we need to do here is basically solve for v final so uh we have two times m g h actually you know what i just realized something we can cross out the ms because the ms are, you know, they're going to get canceled out anyway.

02:34 So now we just have gh plus one half times cosine of 41 times 14 squared.

02:44 And this entire thing is multiplied by 2 is equal to v final squared.

02:50 And so that means v final is equal to the square root of, what is this, g h plus one half, cosine of 41 times 14 squared and all of that times two, that is your final velocity.

03:12 So b wants to know, okay, what if this cliff, this snowball, right? what if it's not launched 41 degrees above the horizontal? what if it's launched 41 degrees below the horizontal? horizontal.

03:29 All right.

03:29 So this is the ball.

03:31 Now let's think about this.

03:32 41 degrees below the horizontal.

03:35 41 degrees.

03:38 Okay.

03:40 Below the horizontal.

03:41 The velocity is still 14 meters per second...