The drawing shows two frictionless inclines that begin at...

The drawing shows two frictionless inclines that begin at ground level $(h=0 \mathrm{m})$ and slope upward at the same angle $\theta .$ One track is longer than the other, however. Identical blocks are projected up each track with the same initial speed $v_{0}$. On the longer track the block slides upward until it reaches a maximum height $H$ above the ground. On the shorter track the block slides upward, flies off the end of the track at a height $H_{1}$ above the ground, and then follows the familiar parabolic trajectory of projectile motion. At the highest point of this trajectory, the block is a height $H_{2}$ above the end of the track. The initial total mechanical energy of each block is the same and is all kinetic energy. The initial speed of each block is $v_{0}=7.00 \mathrm{m} / \mathrm{s},$ and each incline slopes upward at an angle of $\theta=50.0^{\circ} .$ The block on the shorter track leaves the track at a height of $H_{1}=1.25 \mathrm{m}$ above the ground. Find (a) the height $H$ for the block on the longer track and (b) the total height $H_{1}+H_{2}$ for the block on the shorter track.

Key Concepts

Conversion between Kinetic and Potential Energy

This concept explains how a moving object that ascends against gravity converts its kinetic energy into gravitational potential energy. The maximum height reached by an object is determined by the amount of kinetic energy it initially possesses, which is fully transformed into potential energy at the peak of its trajectory when the velocity momentarily becomes zero.

Projectile Motion

Projectile motion occurs when an object is launched into the air and moves under the influence of gravity, following a curved, parabolic path. In cases where an object leaves an inclined plane, its subsequent motion can be analyzed using the principles of projectile motion, particularly to determine its maximum vertical displacement above the point of launch.

Inclined Plane Dynamics

Inclined plane dynamics involves analyzing motion along a slope, where gravitational force causes an object to accelerate or decelerate depending on the angle of incline. This analysis is crucial for determining how much of the kinetic energy of a moving block is converted into height as it ascends an angled surface.

Conservation of Energy

This concept involves the idea that in the absence of non-conservative forces (like friction), the total mechanical energy (the sum of kinetic and potential energy) of a system remains constant. For both blocks on the frictionless tracks, the initial kinetic energy is progressively converted into gravitational potential energy as the blocks move upward, illustrating energy conservation in mechanical systems.

Transcript

00:03 Okay, in this problem, you actually have two problems to worry about.

00:08 So let's just focus on the first one here.

00:12 So what you're actually doing is pushing a block up on this frictionless slope from the bottom to a height of h.

00:22 And you're trying to find what the height h is actually is.

00:31 So, of course, you're going to use the conservation of energy.

00:36 K -e initial plus p -e initial is equal to k -e -final plus p -e final so we have half m v -nodd squared plus m g h is equal to half mvvv squared plus m g capital h we know initially that height is zero on the first slope so therefore the initial potential energy is going to be zero and it's stops at height h, capital h.

01:16 So therefore, the velocity is going to be zero, therefore the k -e final will be zero.

01:23 And now we'll reduce the equation down to this.

01:29 Now, as you can see, that masses is going to be cancelled out.

01:39 And manipulate to isolate h.

01:43 We plug in the given initial velocity of that block and the gravitational constant, i mean the acceleration of gravity, i mean.

01:57 And we would have height of 2 .5 meters off the ground.

02:09 Okay.

02:10 Now let's look at the second case, where we push a block from the bottom of the slope, and then push it up on this frictional slope, and then it would make a parabola.

02:27 From the tip of the slope to make it look like kinematics equation.

02:41 So we could do this in two steps.

02:44 From the bottom to the top of the slope first.

02:49 So we will do the conservation of energy.

02:52 K .e.

02:53 Initial plus p .e.

02:54 Initial equals k .e.

02:55 Final plus p...