Wild geese are known for their lack of manners. One goose is...

Wild geese are known for their lack of manners. One goose is flying northward at a level altitude of $h_{\mathrm{g}}=30.0 \mathrm{~m}$ above a north-south highway, when it sees a car ahead in the distance moving in the southbound lane and decides to deliver (drop) an "egg." The goose is flying at a speed of $v_{\mathrm{g}}=15.0 \mathrm{~m} / \mathrm{s},$ and the car is moving at a speed of $v_{c}=$ $100.0 \mathrm{~km} / \mathrm{h}$ a) Given the details in the figure, where the separation between the goose and the front bumper of the car, $d=$ $104.1 \mathrm{~m},$ is specified at the instant when the goose takes action, will the driver have to wash the windshield after this encounter? (The center of the windshield is $h_{c}=1.00 \mathrm{~m}$ off the ground.) b) If the delivery is completed, what is the relative velocity of the "egg" with respect to the car at the moment of the impact?

Wild geese are known for their lack of manners. One goose is flying northward at a level altitude of $h_{\mathrm{g}}=30.0 \mathrm{~m}$ above a north-south highway, when it sees a car ahead in the distance moving in the southbound lane and decides to deliver (drop) an "egg." The goose is flying at a speed of $v_{\mathrm{g}}=15.0 \mathrm{~m} / \mathrm{s},$ and the car is moving at a speed of $v_{c}=$ $100.0 \mathrm{~km} / \mathrm{h}$
a) Given the details in the figure, where the separation between the goose and the front bumper of the car, $d=$ $104.1 \mathrm{~m},$ is specified at the instant when the goose takes action, will the driver have to wash the windshield after this encounter? (The center of the windshield is $h_{c}=1.00 \mathrm{~m}$ off the ground.)
b) If the delivery is completed, what is the relative velocity of the "egg" with respect to the car at the moment of the impact?

Key Concepts

Two-Dimensional Kinematics

Two-dimensional kinematics involves analyzing motion separately in horizontal and vertical directions using the basic kinematic equations. This separation is useful when dealing with motions where different forces act along different axes, such as in projectile motion where gravity only affects the vertical motion while the horizontal motion remains constant.

Reference Frames

Reference frames are the viewpoints or coordinate systems from which motion is observed and measured. Choosing the appropriate reference frame is essential for correctly applying the laws of physics in dynamics problems, particularly in situations involving relative motion between multiple objects, thereby simplifying the analysis and combination of velocities and displacements.

Projectile Motion

Projectile motion describes the trajectory of an object that is launched into the air and moves under the influence of gravity alone, after its initial velocity is set. In such motion, the horizontal component of velocity remains constant (ignoring air resistance) while the vertical component is uniformly accelerated downward due to gravity. This concept is crucial for analyzing trajectories, determining time of flight, vertical displacement, and impact conditions.

Relative Velocity

Relative velocity refers to the velocity of one object as measured from the reference frame of another moving object. In problems where two objects interact—such as a dropt projectile and a moving vehicle—calculating the relative velocity helps determine the conditions of impact, the approach speed, and eventual collision outcomes.

Transcript

00:01 All right, so it sounds like the goose and the car are going in opposite directions.

00:09 V sub g for the goose is 15 meters per second.

00:17 V sub c for the car is 100 kilometers per hour.

00:30 Well, we're going to have to convert that kilometers per hour to meters per second.

00:34 So, there are 1 ,000 meters in a kilometer, and there's 3 ,600 seconds in an hour.

00:51 So 100 times 1 ,000 divided by 3 ,600, putting that in a calculator, 27 .7 repeating meters per second.

01:12 V sub r, which i'll call the relative velocity, is just, just add those two together.

01:19 I'm going to add 15 to 27 .777, and i get 42 .7 repeating meters per second.

01:34 Now, the height of the goose, which i'm going to write as y sub g, is 30 .0 meters.

02:00 And x, subfinal, where the egg needs to go, is 1 .5 .5.

02:12 104 .1 meters.

02:28 Notice that the goose, that the windshield is one meter off the ground.

02:43 Okay.

02:49 Now the goose is just dropping the egg.

03:03 So v sub y initial is zero.

03:13 So let's do y equals, y initial, which would be y sub g, minus, i mean, plus vy initial t, which is zero, minus one -half g, t squared.

03:49 We also know that the x value is going to be x initial, which i'm going to start as zero, x initial plus v relative times t.

04:28 So we're not concerned about the actual time.

04:37 So what i'm going to do, and i wish i would have set this up so you can see my graph, but i don't believe i did.

04:47 Maybe i did.

04:50 I'm going to pretend that you can see my graph.

04:55 If i go to a graph and calculator like desmos, i can put x in parentheses is vr, which is 42 .7 times t.

05:22 And then y is going to be y sub g, which is 30 minus 9 .000.

05:35 0 .81 over 2 times t squared.

05:47 Okay, now i'm going to drag this over so that i hope you can see it.

05:55 Sharing this tab.

05:57 So unfortunately, you can't see it because i only shared the tab, not the whole screen.

06:10 Nevertheless, when i share it or when i make the graph, i'm not going to share it.

06:19 Draw it for you.

06:21 When i make the graph of these two basically parametric equations, i get, it starts out at 30, and then it goes down like a parabola until it hits the ground, or it reaches zero at, oh man, it's not showing me.

07:03 Don't know why it won't show me the the value where it hits.

07:08 Okay, well, i'm just going in.

07:13 The zero point is 105 .6, comma, zero.

07:32 So, let's think about what i just did.

07:40 Why is y of the goose minus one half gt, and x is that.

07:56 But i want to know where it equals 1.

08:00 And it equals 1, not 0, at 103 .8...