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 $1.40 \mathrm{m}$ from the base of the counter. If the height of the counter is $0.860 \mathrm{m},$ (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

Vector Decomposition

Vector decomposition involves breaking a velocity vector into its horizontal and vertical components using trigonometry. This technique is crucial when determining the resultant velocity’s magnitude and direction at any point during projectile motion, allowing for an analysis of how the object's speed and direction change over time.

Kinematic Equations

Kinematic equations are used to relate displacement, velocity, acceleration, and time for objects in motion. In analyzing projectile motion, these equations are applied independently to both the vertical motion (using gravitational acceleration) and the horizontal motion (with constant velocity) to calculate quantities like time of flight and initial speeds.

Projectile Motion

Projectile motion is the two-dimensional motion of an object that is launched or projected into the air, where the only force acting on it (after launch) is gravity. This concept is essential in solving problems involving objects that follow a curved path due to constant acceleration in the vertical direction, while moving at a constant velocity in the horizontal direction.

Independence of Motion

The principle of independence of motion states that the horizontal and vertical components of an object’s movement are independent of each other. In projectile motion, while gravity affects the vertical motion causing acceleration, the horizontal motion remains constant if air resistance is negligible, allowing for separate analysis of each component.

Transcript

00:01 All right, in this problem, an object slides horizontally over a known height and then lands a known distance away.

00:09 And based just on those two known values, we need to figure out what is the initial velocity that it travels over the edge with.

00:18 All right? we can recognize that the initial velocity in the y direction is zero.

00:23 So if it travels some distance delta y, it's just due to the acceleration.

00:30 In the y direction, and we can write like that.

00:35 All right.

00:36 So in this case, a .y is equal to negative, or i'm sorry, let's say ay is equal to g.

00:45 I'm going to adopt a convention where y and g are both measured positively in the downward direction so that we only need to worry about positive values here.

00:56 Then we can solve for t in this case, and we get t, equals 2 times delta y over g the square root of 2 times delta y over g all right so we can actually take this and plug this directly into a formula to find our v i x at this point if we recognize that there's no initial excel there's no acceleration in the x direction at all then the v i that it has in the initial um its velocity and the initial setup must be its continued horizontal velocity throughout the entire trajectory.

01:34 So we can say that the initial velocity in the x direction is equal to the distance and travels afterward divided by the time it takes to travel that distance.

01:45 And in this case, this will be equal to, oh wait, not equals.

01:56 Okay, perfect.

01:57 So now we can take this and plug in our value for delta x 1 .40 meters plug in our value for delta y 8 or 0 .86 meters and we'll get that v equals 3 .34 meters per second.

02:24 All right.

02:26 The next part of the problem is to, oh sorry, the next part of the problem is to figure out what is the direct of its velocity when it hits the ground.

02:37 So we need to find a theta value for its velocity.

02:42 All right.

02:43 If this is its velocity in the x direction, when it hits the ground, what we need to know at this point is its velocity in the y direction when it hits the ground.

02:51 Let's call that final velocity y.

02:53 All right...

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