When Do Objects Collide and Have the Same Speed?

[jumbogala]

Beginner's Kinematics Problem

Homework Statement

An x-axis likes along an inclined plane, with the positive x-direction down the plane. Two objects A and B travel along the x-axis.

At t=0, object A starts at x=0 with an initial velocity of 0. At t=1s, object B starts at x=10.00 m with a velocity of -5 m/s. Both objects have an acceleration of positive 2 m/s/s.

At what time and position do the objects collide?

At what time do the objects have the same speed?

Homework Equations

x_f= x + v_i(t) + 0.5at²

v_f= v_i+a(t)

The Attempt at a Solution

a) Equate the two equations (because x final is the same) and solve for t.
OBJECT A: x_f= 0 + (0 m/s)(tf-0 s) + 0.5(2 m/s/s)(tf-0)²

OBJECT B: x_f= 10 m + (-5 m/s)(tf-1 s) + 0.5(2 m/s/s)(tf-1)²

I got t= 2.29 seconds using a graphic calculator. Is that correct? Are the equations above correct?

Part b of the question is what's REALLY confusing me. There is no time that I can find that the objects have the same speed.

OBJECT A:
v_f= (0 m/s) + (2.00 m/s/s)(tf-0s)

OBJECT B:

v_f= (-5 m/s/s) + (2.00 m/s/s)(tf-1s)

The two lines, when graphed, never intersect. Also, making object A's equation equal to object B's equation doesn't work either. What am I doing wrong??

 

[LowlyPion]

Homework Statement

An x-axis likes along an inclined plane, with the positive x-direction down the plane. Two objects A and B travel along the x-axis.

At t=0, object A starts at x=0 with an initial velocity of 0. At t=1s, object B starts at x=10.00 m with a velocity of -5 m/s. Both objects have an acceleration of positive 2 m/s/s.

At what time and position do the objects collide?

At what time do the objects have the same speed?

Homework Equations

x_f= x + v_i(t) + 0.5at²

v_f= v_i+a(t)

The Attempt at a Solution

a) Equate the two equations (because x final is the same) and solve for t.
OBJECT A: x_f= 0 + (0 m/s)(tf-0 s) + 0.5(2 m/s/s)(tf-0)²

OBJECT B: x_f= 10 m + (-5 m/s)(tf-1 s) + 0.5(2 m/s/s)(tf-1)²

I got t= 2.29 seconds using a graphic calculator. Is that correct? Are the equations above correct?

Part b of the question is what's REALLY confusing me. There is no time that I can find that the objects have the same speed.

OBJECT A:
v_f= (0 m/s) + (2.00 m/s/s)(tf-0s)

OBJECT B:

v_f= (-5 m/s/s) + (2.00 m/s/s)(tf-1s)

The two lines, when graphed, never intersect. Also, making object A's equation equal to object B's equation doesn't work either. What am I doing wrong??

First of all observe that the equation for the first one released is simply:
x = 1/2*a*t² but it starts a second earlier so, ...
x = 1/2*a*(t+1)²

The second ball then can be described as:
x = 10 -5*t +1/2*a*t²

Since we are in the same time scale (having adjusted the first equation) they will meet when x = x.
Substituting a=2 and solving for time then gives you the solution.
(t+1)² = 10 -5*t + t²

2t + 1 = 10 - 5*t

7*t = 9

Part 2:

v = a*(t+1)

v = 5 - a* t

2*a*t = 3

4 * t = 3

 

[jumbogala]

Haven't I already adjusted the time scale, though, by having the (tf-1 s) part of the second equation?

EDIT: It seems that your solution is exactly one second behind my solution. How do I know which to use?

 

[LowlyPion]

Haven't I already adjusted the time scale, though, by having the (tf-1 s) part of the second equation?

Yes, but I think you must not have been careful along the way.

See the added edit, below for part 2

Also your time scale puts the second ball leaving a second earlier I believe.

 

[jumbogala]

So using (tf-1 s) is not wrong? Because I tried it again and it definitely gives the answer that I originally got.

I think what's happening when you adjust for the time is that you're moving the origin so that x=0 is at x=1s. But the problem, I think is asking for the time that they collide where time was originally 0. So I think my original answer in that case might be correct?

For the second part, I think I understand the first two lines, but not the last two:
2*a*t = 4

4 * t = 4

Shouldn't I be able to equate the first two lines to find the time at which v is equal?

 

[LowlyPion]

So using (tf-1 s) is not wrong? Because I tried it again and it definitely gives the answer that I originally got.

I think what's happening when you adjust for the time is that you're moving the origin so that x=0 is at x=1s. But the problem, I think is asking for the time that they collide where time was originally 0. So I think my original answer in that case might be correct?

For the second part, I think I understand the first two lines, but not the last two:
2*a*t = 4

4 * t = 4

Shouldn't I be able to equate the first two lines to find the time at which v is equal?

Oops. You are right. My way puts the frame in terms of t=0 when the second ball is released. So yes it's 1 second too fast. Sorry for any confusion.

For the a*t terms its the same thing. But yes I set v = v then solved.
Add 1 second to that as well.

 

[jumbogala]

Ah okay, I think I get it now.

Except the a*t terms thing still isn't working for me. Instead of having t+1 on the first equation, it should be just t now, right? So that means on the second equation I will have to have t-1.

This should give:
OBJECT A:
vf= (0 m/s) + (2.00 m/s/s)(tf-0s) or vf=(2.00 m/s/s)(t)

OBJECT B:

vf= (-5 m/s) + (2.00 m/s/s)(tf-1s) or vf=(-5 m/s ) + (2 m/s/s)(t-1)

When equated, there is no solution!

(2t)= -7 + 2t ---> 2t-2t=-7 -----> 0=-7?

 

[LowlyPion]

Ah okay, I think I get it now.

Except the a*t terms thing still isn't working for me. Instead of having t+1 on the first equation, it should be just t now, right? So that means on the second equation I will have to have t-1.

This should give:
OBJECT A:
vf= (0 m/s) + (2.00 m/s/s)(tf-0s) or vf=(2.00 m/s/s)(t)

OBJECT B:

vf= (-5 m/s) + (2.00 m/s/s)(tf-1s) or vf=(-5 m/s ) + (2 m/s/s)(t-1)

When equated, there is no solution!

(2t)= -7 + 2t ---> 2t-2t=-7 -----> 0=-7?

It asks for speed being equal, not velocity.

Note I corrected a small math error in the first way I did it. 4*t = 3

But you add 1 to adjust to the same as this way. Sorry again that my simple math errors may have confused. I was just scratching them out without being careful.

 

[jumbogala]

Oh... I don't know how to calculate speed without giving it a positive or negative sign.

I guess I still don't understand where you got 2*a*t = 4 from, either. Sorry, I'm taking physics for the first time this year and I'm not very good at it yet =)

 

[LowlyPion]

Oh... I don't know how to calculate speed without giving it a positive or negative sign.

On the one side you have v = a*(t) = 2 t

On the other your speed starts out at 5 and slows to deceleration so = 5 - 2*(t-1)

2t = 5 - 2t +2

4 t = 7

 

[jumbogala]

Oh, so when the velocity is -5.00 m/s, that means the speed is 5.00 m/s.

Since the object is slowing down, however, the acceleration could be considered -2 m/s/s.

I get it now!

Thanks for all your help!

 

[LowlyPion]

Oh, so when the velocity is -5.00 m/s, that means the speed is 5.00 m/s.

Since the object is slowing down, however, the acceleration could be considered -2 m/s/s.

I get it now!

Thanks for all your help!

Cheers then.