A dolphin in an aquatic show jumps straight up out of the...

A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m/s. (a) List the knowns in this problem. (b) How high does his body rise above the water? To solve this part, first note that the final velocity is now a known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to his size or orientation.

Key Concepts

Time of Flight in Projectile Motion

Time of flight is the total duration an object spends in the air during projectile motion. In the context of vertical motion, it involves calculating the time to reach maximum height and then doubling it (if the motion is symmetrical) or using kinematic equations directly. This concept is fundamental for understanding how different phases of motion—ascend and descend—contribute to the overall time the object is airborne.

Identifying Known and Unknown Variables

In problem solving, especially in physics, it is crucial to list all provided values (knowns) and clearly state what needs to be determined (unknowns). This organization helps in choosing the most appropriate equations or methods to solve the problem, ensuring that each variable is accounted for and the relationships between them are correctly established.

Unit Analysis and Dimensional Consistency

Unit analysis involves checking that all quantities in an equation are in compatible units, ensuring the mathematical operations lead to correct, meaningful results. It is a critical step for verifying equations and calculations, preventing errors, and confirming that the numerical answer is plausible in the context of the problem.

Vertical Motion Under Constant Acceleration

This concept refers to the study of objects moving in a straight line where the acceleration is constant, typically due to gravity. It involves understanding how an object thrown vertically experiences changes in velocity, displacement, and time as it moves upward and then decelerates under the influence of gravity until it stops momentarily at its peak before descending.

Kinematic Equations

Kinematic equations are mathematical relationships that describe motion involving constant acceleration. Key equations relate displacement, velocity, acceleration, and time. They are essential tools for analyzing problems like vertical jumps or free-fall situations by providing formulas such as v = v0 + at and v² = v0² + 2a(y ? y0), which help determine unknown quantities when given sufficient known variables.

Transcript

00:02 So this problem seems to have not a lot of information in it, but if you kind of understand the situation or analyze the situation correctly, it turns out we actually know four different things about this dolphin.

00:12 My suggestion is if you have a hard time understanding where these numbers come from, is to draw a picture.

00:17 Pictures are super, super helpful when solving kinematics problems, especially if you label the things that you know like velocities and positions and things like that.

00:26 So given this problem, we actually know the initial velocity is 13 .0 meters per seconds.

00:32 That's what we're told.

00:33 We also know that the final position, i'm sorry, the initial position of the dolphin is zero meters.

00:41 We can call the surface of the water zero and just measure from there because that's where the part of this problem is beginning.

00:48 Now, we know the acceleration is due to gravity, so it's negative 9 .8 meters per second squared.

00:53 But we also know at the very peak of this dolphin's jump, at its highest point, it momentarily comes to a stop.

01:01 That's because objects that are thrown upward or that move upward slow down, stop momentarily, then start to speed up in the negative direction as they come back down.

01:11 So at the very, very highest point of anything's travel, it's going to have a velocity of zero meters per second.

01:19 That being said, we can use this information to figure out how high the dolphin jumps.

01:23 So we use all the same information, and the equation that's used is b squared equals initial velocity squared plus 2ax.

01:31 We plug in our numbers, so the final velocity is zero, that's at its highest point, that's equal to 13 .0 squared, and then we add that to 2 times negative 9 .8 times x...

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