Marbles having a mass of 5 g are dropped from rest at A...

Marbles having a mass of 5 g are dropped from rest at $A$ through the smooth glass tube and accumulate in the can at $C .$ Determine the placement $R$ of the can from the end of the tube and the speed at which the marbles fall into the can. Neglect the size of the can.

Key Concepts

Projectile Motion

Once the marble exits the tube, it behaves like a projectile under the influence of gravity. Projectile motion describes the two-dimensional trajectory an object follows when it is launched from a given height with a certain velocity. The vertical motion is influenced by gravity while the horizontal motion continues at a constant speed, which together determine the landing point, allowing calculation of the placement R of the can.

Kinematics Equations

Kinematics equations relate displacement, velocity, acceleration, and time in the analysis of motion. When analyzing the free-fall phase after the marble exits the tube, these equations are used to determine the time of flight (based on vertical displacement) and the horizontal distance traveled (range), which directly informs the proper placement of the can.

Conservation of Energy

This concept involves the transformation of gravitational potential energy into kinetic energy as the marble descends from a height. In a system where friction is negligible (as implied by a smooth tube), the potential energy at the starting point converts completely into kinetic energy at the exit of the tube. This principle provides a way to compute the speed of the marble as it leaves the tube by equating the initial potential energy to the kinetic energy at the exit.

Transcript

00:01 In this problem, we have marbles that travel through a tube and then exit that same tube.

00:05 So we want to find a distance those marbles travel in the x direction so we know where to place a can to catch them.

00:13 And you want to find the velocity as they enter that can.

00:17 So we're given the mass of these marbles, which is 5 grams.

00:23 So you divide by 1 ,000 to get kilograms.

00:25 So you get 0 .05 kilograms.

00:33 So again, we want to find the horizontal distance, and here it's r, and the velocity we'll call velocity c as it hits the can.

00:43 So to find r, or the horizontal distance, we know that we can call it x, and we know that x is equal to the velocity times time.

00:55 So we need to find the velocity and the time.

00:59 So to do that, we use the work energy theorem because we have work being done by the weight of the marbles and it has its own kinetic energy as well.

01:14 So we have the initial kinetic energy plus the work being done.

01:24 And here the work is done from a to p.

01:27 So from the beginning of the tube to the end of the tube is equal to the final kinetic energy.

01:36 We know that initially it has no velocity, so the initial kinetic energy is zero.

01:43 So now we're just left with the work being done.

01:48 And here, the only work being done is by the weight of marble.

01:53 You know that the weight is just mass times gravity.

01:58 And work is force times distance.

02:00 So it's this force, the weight, times the distance, and the distance between the beginning and end of the tube is three, the top of the top.

02:08 Of the tube minus two the bottom of the tube.

02:13 This is equal to the final kinetic energy, which is one half mass of the marble times v.

02:20 We'll call it vb squared.

02:22 So this is the velocity that the marbles exit the tube with.

02:26 And this is what we want to solve for.

02:29 So plug in your numbers.

02:33 We got, so for the left side, we have .005 times gravity 9 .8 times...

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