As a runaway scientific balloon ascends at 19.6 m / s, one of its instrument packages breaks fr

As a runaway scientific balloon ascends at 19.6 m / s, one...

As a runaway scientific balloon ascends at $19.6 \mathrm{~m} / \mathrm{s}$, one of its instrument packages breaks free of a harness and free-falls. Figure $2-34$ gives the vertical velocity of the package versus time, from before it breaks free to when it reaches the ground. (a) What maximum height above the break-free point does it rise? (b) How high is the break-free point above the ground?

Key Concepts

Velocity-Time Graph Analysis

Velocity-time graphs are useful tools for visualizing and analyzing motion. The graph enables one to identify moments such as the peak of the trajectory when velocity becomes zero. By interpreting the slopes and areas under the curve, one can deduce key kinematic parameters like displacement or maximum height, making it an essential tool in studying vertical motion.

Free Fall and Acceleration Due to Gravity

Free fall is a specific type of motion where gravity is the only force acting on an object. The acceleration due to gravity is nearly constant near the Earth's surface and is a critical factor in calculating how fast the object accelerates downward, as well as in determining how high it can ascend when thrown upward in a case of initial upward motion.

Kinematic Equations

Kinematic equations are used to relate displacement, velocity, acceleration, and time for an object moving with constant acceleration. They form the backbone of solving problems involving vertical motion, allowing one to determine maximum height, time of flight, or final velocity by setting up equations based on the object's initial conditions and the effect of gravity.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject only to gravity, making its path a parabola. In the vertical component of projectile motion, analysis often involves determining key points such as when the velocity becomes zero, which marks the maximum height of the trajectory.

Transcript

00:01 All right, for this problem, we have a diagram of velocity versus time.

00:08 It goes out to eight seconds.

00:13 And let's see, it looks like over, then down.

00:24 But i've put my t -axis in the wrong place because it's zero about halfway through.

00:37 So, this is four.

00:41 This is eight.

00:43 The peak is at two.

00:47 All right.

00:49 So let's see.

00:55 Initially, it's at 19 .6 meters per second, which seems awfully fast for a balloon.

01:13 Then, wait.

01:18 Okay, something breaks free from the balloon and free falls.

01:40 Okay, so a, we want the height above the break free point.

01:48 Well, v squared equals v initial squared minus 2a delta y.

02:01 So delta y is going to be v squared, which is zero at the maximum height at the peak.

02:14 So it's going to be v0 squared over 2a, where a is is g, gravity.

02:23 Okay, so v0 squared is 19 .6 and g is 9 .81.

02:38 So v squared over 2g, i'm putting this into a calculator, would be 19 .6 meters.

02:56 B, how high is the break -free point above the ground? well, um, um, okay, let's read this again.

03:31 It says to when it, it's to when, but then it says from, oh, from before it breaks free to when it reaches the ground.

03:43 So what we know is it reaches the ground right here.

03:46 And it's been decelerating for six seconds.

03:50 So how high is that point? how high is the break free point? okay.

04:00 So, y equals y initial plus vy initial t minus t minus one half g t squared.

04:12 But i'm calculating from the peak...

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