Find the time of a certain velocity

[pberardi]

Homework Statement

A time-dependent force, F = 8i – 4tj N, is exerted on a 2.00 kg object initially at rest. a) At what time will the object be moving with a speed of 15 m/s? b) How far is the object from its initial position when its speed is 15 m/s? c) Through what total displacement has the object traveled at this moment?

Homework Equations

F = ma
vf = v0 + at

The Attempt at a Solution

I took the mag of the F which is sq(80) = ma so: 8.9 = 2a, therefore a should be 4.47.
Then I used vf = vo + at. It is at rest so vo = 0 so: 15 = 4.47t, t = 3.35 and it is incorrect. Can someone please tell me what I did wrong? thanks.

 

[Doc Al]

Since the force, and thus the acceleration, is time-dependent, you can't use formulas that only apply for constant acceleration. Hint: Use a bit of calculus.

 

[pberardi]

What I did was integrate the force vector to get <8t, -2t^2> = 2a but I still have two unknowns.

 

[Doc Al]

What I did was integrate the force vector to get <8t, -2t^2> = 2a but I still have two unknowns.

(You mean = 2v.) The only unknown is the time. (Find the magnitude of that velocity vector.)

 

[pberardi]

Ok tell me if this will work: F = ma so a = F/m so a = <4, -2t>. v = <4t, -t^2> but speed is the mag of velocity so 15 = sq(16t^2 + t^4). So 225 = 16t^2 + t^4 which gives me four roots one of them being 3. Is this good?

 

[Doc Al]

Perfecto!

 

[pberardi]

Great thanks. But now I am having a bit of trouble with the position. It is asking me for the vector here so I split it up into components.
For x I get xf = xi + vxit + 1/2(ax)t^2 which is xf = 0 + 4t(t) + (1/2)4t^2 at t = 3 gives 54
For y I get yf = yi +vyit + 1/2(ay)t^2 which is yf = 0 + (-t^2)t + (1/2)(-2t)t^2 at t = 3 is -54. They give a number for the answer so I should mag the vector and I get 76.4 but it is incorrect. Please?

 

[Doc Al]

For x I get xf = xi + vxit + 1/2(ax)t^2

Same issue as before: Don't try to apply constant-acceleration formulas when the acceleration is not constant. Same advice as before: Use calculus.

 

[pberardi]

But I have already done the calculus. I am just using the v and a that I had before and just plugging them in. I really don't understand why this doesn't work.

 

[Doc Al]

But I have already done the calculus.

You used calculus to find the velocity; now use it to find the displacement.

I am just using the v and a that I had before and just plugging them in. I really don't understand why this doesn't work.

Because the acceleration is not a constant over the interval from t = 0 to t = 3 seconds. So those kinematic formulas do not apply.

 

[pberardi]

I do not know any other formulas that give me a position. Could you give me a little bigger hint?

 

[Doc Al]

Sure. Just like a = dv/dt, v = dx/dt.

 

[pberardi]

Thanks. Just curious. Since the kinematics do not apply, what would we do if we were given an initial velocity and an initial position?

 

[Doc Al]

Whenever you integrate, you'll have constants of integration. The initial conditions will determine those constants.

For example, since the object is initially at rest, you know that at t = 0, v = 0. And since all we care about are displacements from the initial position, we can say that at t = 0, the object is at x = y = 0.

 

[Picapichu]

For the first part how do you get 225 = 16t^2 + t^4?

 

[Doc Al]

For the first part how do you get 225 = 16t^2 + t^4?

See post #5. To go from a to v, integrate.