A key falls from a bridge that is 45 m above the water. It falls directly into a model boat, mo

step 1 So we've got a key falling from a bridge. Y0 is 45 meters and V0 is zero, because it's basically

A key falls from a bridge that is 45 m above the water. It...

A key falls from a bridge that is $45 \mathrm{~m}$ above the water. It falls directly into a model boat, moving with constant velocity, that is $12 \mathrm{~m}$ from the point of impact when the key is released. What is the speed of the boat?

Key Concepts

Decoupling of Vertical and Horizontal Components

In projectile motion, the vertical and horizontal components of motion are independent of each other. The vertical motion under gravity does not affect the horizontal motion with constant velocity, which allows one to compute the time of free-fall independently and then use that time to determine the horizontal displacement or speed.

Constant Velocity Motion

Constant velocity motion refers to movement where an object covers equal distances in equal intervals of time, without acceleration. This concept is directly applied when calculating how fast the boat must move to cover a given horizontal distance in the time it takes the key to fall, implying there is no change in the boat’s speed during its motion.

Kinematic Equations for Uniform Acceleration

The kinematic equations relate displacement, initial velocity, acceleration, and time for objects moving with constant acceleration. In the context of free-fall, these equations are used to determine how long it takes for an object to fall a specified distance under the acceleration due to gravity.

Free-Fall Motion

Free-fall motion describes the movement of an object solely under the influence of gravity, without any initial vertical velocity and ignoring air resistance. When an object falls from a certain height, its motion is dictated by a uniform acceleration due to gravity, allowing the computation of the fall time using the relation between distance, gravitational acceleration, and time.

Transcript

00:01 So we've got a key falling from a bridge.

00:04 Y0 is 45 meters and v0 is zero, because it's basically just being dropped.

00:22 It falls onto a boat, but it's 12 meters from the point of impact.

00:33 It's 12 meters from the point of impact.

00:39 It falls onto a boat, 12 meters.

00:46 From the point of impact, which i guess is from the water.

00:50 So i think it's telling us that there's a boat on the water, and it's 12 meters here.

00:59 I think.

01:07 No.

01:09 No, no, no, no.

01:10 That's not right.

01:12 It falls into a boat that's moving.

01:19 Okay.

01:20 So this is a two -dimensional problem.

01:22 Basically, there's the boat here.

01:23 And it's moving and it is 12 meters away when the key up here is released.

01:38 Okay.

01:40 What's the speed of the boat? got it.

01:44 So let's look at the key y equals y zero plus v0 t minus one half g t squared this is zero um this is zero so 45 equals one half gt squared so t is going to be the square root 45 times 2 is 90 of 90 g so putting that in a calculator t equals square root of 90g.

02:33 Huh? 29 .7 seconds? seems like a long time.

02:44 So let's think about this again...

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