The Bellagio Hotel in Las Vegas, Nevada, is well known for...

The Bellagio Hotel in Las Vegas, Nevada, is well known for its Musical Fountains, which use 192 HyperShooters to fire water hundreds of feet into the air to the rhythm of music. One of the HyperShooters fires water straight upward to a height of $240 \mathrm{ft}$. a) What is the initial speed of the water? b) What is the speed of the water when it is at half this height on its way down? c) How long will it take for the water to fall back to its original height from half its maximum height?

The Bellagio Hotel in Las Vegas, Nevada, is well known for its Musical Fountains, which use 192 HyperShooters to fire water hundreds of feet into the air to the rhythm of music. One of the HyperShooters fires water straight upward to a height of $240 \mathrm{ft}$.
a) What is the initial speed of the water?
b) What is the speed of the water when it is at half this height on its way down?
c) How long will it take for the water to fall back to its original height from half its maximum height?

Key Concepts

Time Symmetry in Projectile Motion

In ideal projectile motion, the time taken to ascend to a certain height is equal to the time taken to descend from that height back to the starting level, assuming the launch and landing heights are the same. This symmetry is useful in calculating the duration of different segments of the motion, such as the time it takes for an object to fall from a given height.

Projectile Motion

Projectile motion refers to the motion of objects that are launched into the air and then move under the influence of gravity alone, following a parabolic path. This concept is central to analyzing trajectories, maximum heights, velocities, and time intervals, regardless of the presence of air resistance in basic physics problems.

Kinematic Equations

Kinematic equations are a set of formulas that relate displacement, initial velocity, final velocity, acceleration, and time in uniformly accelerated motion. They are used to calculate unknown parameters such as initial speeds, velocities at various heights, and time durations during the motion.

Acceleration Due to Gravity

Acceleration due to gravity is the constant rate at which objects accelerate when falling freely near the Earth's surface. This downward acceleration is a key factor in projectile motion, as it dictates how quickly the speed increases on the downward path and how it impacts the time to reach a given height.

Energy Conservation in Vertical Motion

The conservation of energy principle states that in the absence of non-conservative forces, the total mechanical energy (kinetic plus potential) remains constant. This concept is used to relate the initial kinetic energy of a projectile with its gravitational potential energy at its maximum height, which can also help in calculating speeds at various points in the trajectory.

Transcript

00:01 And in this problem here, we have a water jet, which is firing from the ground here in a fountain.

00:08 Let's say, call that zero, which is zero feet.

00:14 And the water launches up into the air up to xm for max of 240 feet.

00:23 And we have a few questions here regarding this water.

00:27 And what we do know is that we only have one force acting on the water.

00:31 And that is gravity, which is, we'll say that the force is minus g, because gravity acts downwards.

00:42 So we have a, let's say, x, x that goes like this.

00:50 And we have three questions here.

00:54 We have a.

00:56 What is the initial velocity of the water? b, what is the velocity when it's falling down? down at half of its maximum value and see later than what is the time delta t to fall from x and half on the way down to zero.

01:23 So we'll start with a like this and we have some boundary conditions which need to be satisfied.

01:36 We do know that when we go from the bottom up to the top, we have initially, we're initially at x0 equals zero.

01:52 We're going to reach a maximum height of 240 feet.

01:58 We start with the velocity v nod, which we don't know.

02:04 We find it and we finish the velocity of.

02:08 Of zero.

02:10 So on the top of our fountain here, we have zero velocity.

02:19 And we're going to use newton's laws to find these values.

02:24 And as you know, we have f equals an a, but you can also just write this.

02:37 We know that we have a equals d2.

02:45 X d t squared and from integrating this expression we will find a few different equations the first one is we can say is v equals a t plus v zero and the second one is x equals if you integrate the above equation one more time a t squared over 2 plus b0 t plus x not.

03:25 And important note here, because we only have one acceleration which is acting on our water, we have a equals minus g.

03:39 Okay, so we're going to use these equations...

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