Calculating Velocity and Acceleration of Ball from Child A

In summary, the problem involves a child on a Merry Go Round who is running along the edge at 0.4m/s, while another child throws a ball aimed at the center of the Merry Go Round. The ball has a velocity of 10m/s and the Merry Go Round is turning at an angular velocity of 0.5 rad/s. To find the velocity and acceleration of the ball from the perspective of the running child, we need to consider the superposition of their motions. This can be calculated by adding the angular velocity of the Merry Go Round to the child's angular velocity, and then using that to find the time it takes for the child to make one rotation. From there, we can use the equations for angular
  • #1
physicsone
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Homework Statement


A child A is on a Merry Go Round that is turning at an angular velocity of 0.5 rad/s in the
clockwise direction. A ball B thrown from another child who is 4.5 m from the center of the MGR has a velocity of 10m/s and directly aimed at the center of the MGR as shown in the figure. Now, let us assume that the child A is running along the edge of the MGR at 0.4m/s in the clockwise direction with respect to the MGR. Find the velocity and acceleration of the ball seen from the child A.


Homework Equations



I know, W,A = W,MGR + W,A/MGR where W=angular velocity

And i belive, W,A/MGR = 2pi/(time for A to make one rotation)

The Attempt at a Solution



I am not sure how to calculate the time for A to make one rotation
 
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  • #2
To me this a very intersting problem, and have no knowledge whatsoever from experience on similar problems, so likely of no use. Also I see no picture nor even understand what you have posted.
My limited undertanding:

first, the ball B will hit the center target. Thrower is not in motion nor is the center of the MGR. Nor is the ang velocity of the MGR important, or our A child, for that matter except insofar as we have a total ang velocity. So the perspective from child a is really a superposition of his motion, the MGR's and the balls. The first two add in simple way.

so if we could generate a superposition of spatial coordinates as a function of time for both ball and child, then we could accurately describe motion as seen by the child?

Then if childs place in space C(x,y)= R*(cos(wt) + sin(wt)) and the ball along the x coordinate as 4.5-10*t, then there should be a way to portray the combined motions. But this is where I get stuck, not knowing initial x,y of child how to proceed? Obviously at 12o'clock looks much different to observer than at 6. But in general,

Vx=d(C(x,y))/dt+10,
Vy=d(C(x,y)/dt (these are partials obviously) so knowing both x' and x", y' and y" we can map out perceived trajectory. I very much doubt this cumbersome notation is what is asked for, just wanting to understand, and I still don't get it, for instance

If I were to translate these to words, I would think along the x-axis the ball either accelerates briefly, then still (depending on R and w vs 10) and then slows or vice versa, while on the Y depending on which starting point I chose, say 12noon, the overall motion would be like that of a heat seeking missile and assuming i could make the 1/4 turn before getting it between the eyeballs,would then veer to the left and slow down. There must be a formal way of mixing these frames of reference? Help.
 
  • #3
, but I will assume it takes 10 seconds for simplicity. Therefore, W,A/MGR = 2pi/10 = 0.2 rad/s.

To calculate the velocity of the ball seen from child A, we can use the formula v = rω, where v is the tangential velocity, r is the distance from the center of rotation, and ω is the angular velocity. In this case, the distance from the center of rotation for child A is 4.5m + 0.4m = 4.9m. Therefore, the velocity of the ball seen from child A is v = (4.9m)(0.5 rad/s) = 2.45 m/s.

To calculate the acceleration of the ball seen from child A, we can use the formula a = rα, where a is the tangential acceleration, r is the distance from the center of rotation, and α is the angular acceleration. In this case, the distance from the center of rotation for child A is still 4.9m. We can calculate the angular acceleration using the formula α = Δω/Δt, where Δω is the change in angular velocity and Δt is the change in time. In this case, the angular acceleration is 0.5 rad/s / 10s = 0.05 rad/s². Therefore, the acceleration of the ball seen from child A is a = (4.9m)(0.05 rad/s²) = 0.245 m/s².

In summary, the velocity of the ball seen from child A is 2.45 m/s and the acceleration of the ball seen from child A is 0.245 m/s². These values show that the ball is moving at a constant speed and experiencing a constant acceleration in the direction of the Merry Go Round's rotation.
 

1. How do I calculate the velocity of a ball thrown by Child A?

The velocity of a ball can be calculated using the formula v = d/t, where v is the velocity, d is the distance traveled, and t is the time taken. To calculate the velocity of a ball thrown by Child A, you will need to measure the distance the ball traveled and the time it took to travel that distance.

2. What is the difference between velocity and speed?

Velocity and speed are often used interchangeably, but they are not the same. Velocity is a vector quantity that includes both magnitude (speed) and direction. Speed, on the other hand, is a scalar quantity that only includes magnitude. In the context of a ball thrown by Child A, velocity would include both the speed of the ball and the direction in which it was thrown, while speed would only refer to how fast the ball is moving.

3. How do I calculate the acceleration of a ball thrown by Child A?

The acceleration of a ball can be calculated using the formula a = (vf - vi)/t, where a is the acceleration, vf is the final velocity, vi is the initial velocity, and t is the time taken. To calculate the acceleration of a ball thrown by Child A, you will need to measure the initial and final velocities of the ball, as well as the time it took to change from one velocity to the other.

4. Can I use the same formula to calculate the velocity and acceleration of any object?

Yes, the formulas for calculating velocity and acceleration are universal and can be used for any object in motion, including a ball thrown by Child A. However, it is important to note that the initial and final velocities measured may vary depending on the object's mass, shape, and other factors.

5. How can I use the calculated velocity and acceleration of a ball thrown by Child A in scientific experiments?

The calculated velocity and acceleration of a ball thrown by Child A can be used in various scientific experiments and studies. For example, it can be used to analyze the ball's trajectory, the force applied by Child A in throwing the ball, and the effects of air resistance on the ball's motion. It can also be used to compare the throwing abilities of different children or to study the relationship between velocity, acceleration, and distance traveled.

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