You serve a tennis ball from a height of 1.8 m above the ground. The ball leaves your racket wi

step 1 All right. Tennis is one of my favorite sports. So that's nice to have a tennis question. I don'

You serve a tennis ball from a height of 1.8 m above the...

You serve a tennis ball from a height of $1.8 \mathrm{~m}$ above the ground. The ball leaves your racket with a speed of $18.0 \mathrm{~m} / \mathrm{s}$ at an angle of $7.00^{\circ}$ above the horizontal. The horizontal distance from the court's baseline to the net is $11.83 \mathrm{~m},$ and the net is $1.07 \mathrm{~m}$ high. Neglect spin imparted on the ball as well as air resistance effects. Does the ball clear the net? If yes, by how much? If not, by how much did it miss?

Key Concepts

Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. It involves independent horizontal and vertical motions, which are analyzed separately to understand the object's trajectory.

Velocity Decomposition

Velocity decomposition involves breaking the initial velocity vector into horizontal and vertical components using trigonometric functions. This separation is essential because the horizontal motion remains uniform while the vertical motion is influenced by gravity.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. They are used to calculate position, velocity, and time for both horizontal and vertical components of projectile motion, allowing the determination of an object's position at any given time.

Time Determination

Time determination in the context of projectile motion involves using the horizontal motion, where the time to reach a particular horizontal distance is calculated by dividing that distance by the horizontal component of the initial velocity. This time is then used in vertical motion calculations.

Clearance Evaluation

Clearance evaluation refers to comparing the height of a projectile at a specific horizontal distance with the height of an obstacle. This analysis determines whether the projectile clears the obstacle and, if so, by how much, or by how much it falls short.

Transcript

00:01 All right.

00:01 Tennis is one of my favorite sports.

00:05 So that's nice to have a tennis question.

00:10 I don't think i've ever actually seen a tennis question in a physics book.

00:15 So the tennis ball leaves from a height of 1 .8 meters above the ground.

00:21 So there you are playing tennis.

00:28 Okay, so up here is 1 .8 meters.

00:34 So, y initial is 1 .8 meters.

00:47 V initial is 18 meters per second, and the angle is 7 degrees above the horizontal.

01:10 So, the distance to the net is 11 .3 meters.

01:27 And the height of the net is 1 .07 meters.

01:47 So i'm going to show this going over and clearing the net.

01:59 Okay.

02:01 So we need to figure out when does...

02:11 Okay, when it travels 11 .3 meters, what's the height? so i'm going to use x.

02:23 Equals x initial, which is zero, plus v initial in the x direction, which would be cosine, theta, times t.

02:44 Okay? so that's going to tell me when it reaches the net.

02:52 So t is going to be x over v0 cosine theta now what will be the y value at that time y is going to be y initial which we know plus v y t minus one half g t squared so g is 9 .81 meters per second squared.

03:40 Okay.

03:42 So, y is going to be y0, which we know, plus v0 sine theta, but t is x over v0 cosine theta, minus one half g.

04:12 X over v0 cosine theta squared okay i see that these two v zeros cancel out and so i've got y equals y0 plus x which would be x of the net i should have been writing x sub n x sub n x sub n sign over is tangent theta minus one -half g and there's really nothing i can do with this squared.

05:15 Okay, so let me put that into a calculator...

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