# Distance traveled by particle

1. Jun 29, 2015

### takando12

1. The problem statement, all variables and given/known data
The position of a particle along x-axis at time 't' is given by x=1+t-t2.The distance travelled by the particle in the first second is.
A) 1m B) 2m C) 2.5m D) 3m

2. Relevant equations

3. The attempt at a solution
Differentiating x wrt t we get, v=1-2t and double differentiating x wrt to t we get a=-2m/s2.
I plotted a V-t graph. The velocity will be 0 at t=0.5 sec , 1 at t=0sec and -1 at t=1 sec. Finding the area under the graph I got 0.5m. Where have I gone wrong? I got 2.5m when I substituted t=0.5 in x and multiplied it by 2. But why is the graph method not working?

2. Jun 29, 2015

### Nathanael

That's more complicated than it needs to be.

The function x(t) already gives you the position at each time, so there's no need to differentiate.

How did you find the area under the curve? By integrating the velocity w.r.t. time? That just gets you back to the original function (except now you lost information about the initial position). Why differentiate just to integrate? It's redundant (and leaves more room for mistakes).

3. Jun 29, 2015

### takando12

I differentiated it to find out when the velocity will be 0. Only after knowing that can i split the time period and use the x(t) to find the distance.
I drew a v-t graph and there were two triangles , i added their area using 1/2*b*h*2.
[1/2*1/2*1] + [1/2*1/2*1]=0.5 m.

4. Jun 29, 2015

### haruspex

Good so far.
Good again.
It's the 2.5m answer that's wrong. The question asks for the distance travelled; the particle didn't start at 0.

5. Jun 29, 2015

### takando12

But 0.5 is not even in the options. And the answer key says it's 2.5m. Just to be clear, when x(t) =1 for t=0 sec. So when I calculate for 0.5 sec, the value of x I get is inclusive of the 1m in the beginning? So I should subtract x(0.5)-1= 1.25-1=0.25. And so the final answer 0.25*2=0.5 fits with the v-t graph.
So the options are wrong?

6. Jun 29, 2015

### Nathanael

Sorry about that, I misinterpreted the question. (I thought it was asking for the displacement.)

Yes, the options are wrong, good job.

(As Haruspex pointed out, the answer is probably meant to be 2.5m, but that is a mistake. Your answer is correct.)

7. Jun 30, 2015

### takando12

Thank you so much sir or rather sirs.