Two children are playing on two trampolines. The first child...

Two children are playing on two trampolines. The first child bounces up one-and-a-half times higher than the second child. The initial speed up of the second child is 4.0 m/s. ($a$) Find the maximum height the second child reaches. ($b$) What is the initial speed of the first child? ($c$) How long was the first child in the air?

Key Concepts

Kinematics Under Constant Acceleration

This concept involves analyzing motion with constant acceleration, such as gravity, using formulas that relate displacement, initial velocity, time, and acceleration. It is essential for solving problems in vertical motion by applying equations like v² = u² + 2as and s = ut + (1/2)at².

Maximum Height in Vertical Motion

The maximum height reached by an object projected upward is determined by the point where its velocity becomes zero. Using the kinematic equations, one can derive the height reached as a function of the initial velocity and gravitational acceleration, typically as h = u²/(2g).

Time of Flight for Upward Motion

For vertical motion under gravity, the time taken for an object to reach its highest point is found by setting the final velocity to zero and solving the equation v = u + at. This helps in calculating the overall duration of the motion as well as the time intervals for specific segments of the trajectory.

Scaling and Proportional Relationships in Kinematics

Understanding how changes in initial speed affect the maximum height or time of flight is crucial. Since the maximum height is proportional to the square of the initial velocity, any scaling in the initial velocity leads to a quadratic change in the height reached, highlighting the importance of proportionality in kinematic relationships.

Transcript

00:01 Okay, so today we're doing kinematics problem.

00:03 The basics of the problem is that we have two people jumping on a trampoline.

00:06 The first guy jumps one and a half times the height of the second guy, and the second guy leaves at four meters per second, and then we're solving these three problems.

00:17 So how i like to solve a kinematics problem is simply create this sort of chart here, and this will help get your thoughts in a line and figure out what you're solving for.

00:27 So in this case, for problem a, we're solving for person two's jump.

00:30 And we have to choose an initial and final state that we want to use in these equations.

00:36 It's good to choose states you know something about.

00:38 So let's call the initial when they leave the trampoline because we know how fast they leave the trampoline at.

00:43 And since we're trying to find information about how high they jump, let's talk about the apex being the final.

00:49 So these are two important states in the system.

00:53 So the difference between these systems, delta y is what we're trying to find.

00:57 Vi, we know is 4 meters per second from the problem statement.

01:01 At the apex, we know that they have no velocity and gravity in freefall is negative 9 .8 meters per second squared.

01:10 Now, since we have three values in this chart, that means this is a kinematics problem that allows us to solve for the other two values.

01:18 So to solve for delta y, the easiest one to use is this one right here, vf squared equals vi squared plus 2a delta y.

01:26 Right, simply plug in zero here, and you get four square.

01:31 Is equal to 2 plus 2 times negative 9 .8 times delta y.

01:37 And if you solve for delta y, you get 0 .816 meters.

01:42 So that's our solution to problem a.

01:46 Similarly, let's talk about the second one.

01:50 In this case, we're talking about person 1.

01:53 We know their initial states and final states.

01:55 Since we know stuff about how high they jumped, it's probably a good idea to keep the same as before.

02:03 We know that person 1 jumped 1 .5 times person 2, so we simply multiply this by 1 .5, and you get 1 .24.

02:15 We don't know v -i, that's what we're trying to find.

02:18 We know vf is 0 at the apex, and we know ay is negative 9 .8 meters per second squared...

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