In a local bar, a customer slides an empty beer mug down the...

In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momentarily distracted and does not see the mug, which slides off the counter and strikes the floor at distance $d$ from the base of the counter. If the height of the counter is $h$, (a) with what velocity did the mug leave the counter and (b) what was the direction of the mug's velocity just before it hit the floor?

In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momentarily distracted and does not see the mug, which slides off the counter and strikes the floor at distance $d$ from the base of the counter. If the height of the counter is $h$,
(a) with what velocity did the mug leave the counter and (b) what was the direction of the mug's velocity just before it hit the floor?

Key Concepts

Projectile Motion

Projectile motion refers to the movement of an object that is launched into the air and moves under the influence of gravity. The motion is typically analyzed as two independent components: a horizontal component with constant velocity and a vertical component with uniformly accelerated motion due to gravity. This concept helps in understanding how objects travel through space when launched from an elevated position.

Separation of Motion into Components

When dealing with projectile motion, it is crucial to break the object's initial velocity into horizontal and vertical components. This separation allows each component to be analyzed independently using appropriate kinematic equations: the horizontal motion remains at constant speed, while the vertical motion experiences acceleration due to gravity. This technique is fundamental in solving for time of flight, range, and final velocity direction.

Kinematic Equations

Kinematic equations provide the mathematical framework for relating displacement, velocity, acceleration, and time in uniformly accelerated motion. For vertical motion under gravity, these equations are used to calculate the time it takes for an object to fall from a given height. Similarly, they assist in determining the vertical velocity component at any point during the fall, which is necessary for analyzing the overall motion.

Determining Velocity Magnitude and Direction

In solving problems involving projectile motion, calculating the magnitude and direction of the velocity at any point requires combining the horizontal and vertical components vectorially. The initial horizontal velocity can be determined from the horizontal distance traveled and the time of fall, while the vertical component at impact is obtained from the acceleration due to gravity. The overall velocity’s direction is then found by computing the angle of the resultant vector using trigonometric functions.

Transcript

00:01 All right, for this problem, an object moves horizontally off of some height h and lands some distance d away.

00:07 And based only on the distance d and the height h, we have to figure out what is its initial velocity in the x direction before it leaves the height, before it leaves the cliff or whatever.

00:18 The initial velocity in the y direction we have to recognize is zero because it is initially not traveling up or downward at all.

00:27 And the acceleration in the right direction we know is due to gravity, and we know that there's no acceleration in the x direction because this is just a projectile problem.

00:39 Based on this piece of information, we can set the initial velocity in the x direction to some constant value.

00:47 We know that if it's moving horizontally, it's going to be moving with a horizontal velocity that's constant.

00:53 And so we can set this equal to horizontal distance traveled divided by t the time it takes to fall.

01:00 All right.

01:00 Now, we don't know t, but we can figure out t based on vertical motion.

01:07 We know that if it travels some distance y, that's going to be equal to vyt plus one -half a y -t squared.

01:20 This is zero because it's not moving upward and downward at all.

01:24 And so what we get is our height, and we're just going to use positive values.

01:28 Let's say if it travels in the positive direction due to a positive acceleration, we don't have to worry about anything.

01:35 We get something that looks like this.

01:36 The height fallen is equal to one half times gt squared.

01:40 And this can be solved for t based on quantities that we know.

01:46 All right.

01:47 So whether g or h is defined as positive or negative, this whole thing has to be positive on the inside or else the square root won't work out.

01:55 All right, if we plug this into our formula for our x, what we get is something that looks like this.

02:06 D times the square root g over 2h.

02:13 All right.

02:15 And that is, that is the answer...

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