How Does Acceleration Affect Motion in the XY Plane?

AI Thread Summary
The discussion focuses on a particle moving in the XY plane with an initial velocity and only an x-component of acceleration. The components of the velocity vector are derived, showing that the total velocity at any time is influenced by both initial velocity and acceleration. At t=5 seconds, the particle's position is calculated, yielding coordinates of (150, -75) and a displacement magnitude of 170 meters. The conversation shifts to determining the distance traveled, emphasizing the need to integrate the velocity function over time rather than simply adding vector components. Ultimately, the correct approach involves finding the integral of the speed function to accurately calculate the total distance traveled.
mohdhm
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1)
A particle starts form the origin at t=0 with an initial velocity having an x compoent and a y component. the particle moves in the xy plane with an x component of acceleration only


a)
Determine the components of velocity vector at any time and the total velocity vector at any time
Given Info: Vxi=20m/s, Vyi=-15m/s, Ax=4.0m/s^2, Ay=0.
Solved.
Answer: Vf=Vxi+Vyi = [(20+4.0t)(i hat symbol?) - 15(j hat symbol?)]m/s

b) calculate the velocity and speed of the particle at t=5?

Ok the tricky part... both velocity and speed have the same magnitude because with 2 dimensions you have to use the pythagorean theorm to add the magnitude of the x and y components. I also concluded that vector = speed but with theta(direction).

solved the problem for part B.

c) determine the x and y coordinate of the particle at any time t and the position vector of this time.

xf=vit=1/2at^2= (20t+2.0t^2)m
yf=vit=(-15t)m

so the position vector at any time t is [(20t+2.0t^2)i -15tj]m

Then the example question says.. let's test it out. and gives me some values

at t=5, x=150, y=-75

what is the magnitude of the displacement?

rf=sqrt[(150^2+(-75)^2]=170m

"note that this isnot the distance that the particle travels in this time! Can you determine the distance from the available data?

Now how do you determine the distance at t=5 ? do you just add the vector components of distance? If someone can please answer that last question, i could rest my mind :eek:
 
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mohdhm said:
Now how do you determine the distance at t=5 ? do you just add the vector components of distance?
I assume you mean add the components of the displacement. No, that won't do. Hint: Find an expression for the speed of the particle as a function of time.
 
so you find the intergral of Vx and Vy and add them up?

d(t)=Velocity * integral of 0 to 5 where f(x) is dt ?
 
Find an expression for the speed of the particle and integrate that from t = 0 to 5 seconds. What's the speed as a function of time?
 
finding the integral of vf=vi+at gives me d= vit+1/2at^2 and we just solve from there... using the absolute value...
i integrated the vf=vi+at and got the expected d=vit+1/2at^2 which gives me DISPLACEMENT

What is the distance, what is the solution of this problem.
 
Last edited:
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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