# Particle Motion in Space

1. Jun 17, 2015

### RyanH42

1. The problem statement, all variables and given/known data
A particle moves so that its equation of motion in vector form given is given by $\vec{R}=((sint^-1)/2+t/2√(1-t^2))\vec{i}+1/2t^2\vec{j}$ , $0≤t<1$
a)Show that particle moves with a constant speed.
b)Compute $\vec{v}$ and $\vec{a}$,and verify that $\vec{v}$.$\vec{a}$=0 (dot product).(As it should be when speed is constant
c)Since the magnitude of speed is constant ,must the magnitude of acceleration also be constant ?

2. Relevant equations
$\vec{v}=d\vec{R}/dt$
$\vec{a}=d\vec{v}/dt$

3. The attempt at a solution
For a), I have to do $\vec{v}=d\vec{R}/dt$ and then I have to do this $\|\vec{v}\|$.If $\|\vec{v}\|$ this do not contain t it means $\|\vec{v}\|$ is constant
For c)$\vec{v}$.$\vec{a}$=0 this means $\|\vec{v}\|.\|\vec{a}\|.cosθ=0$ we know that $\|\vec{v}\|$ is not zero so theres two option 1) $\|\vec{a}\|$ will be zero or $cosθ$ will be $0$.If $cosθ$ is zero then theres no need to be constant magnitude of $\vec{a}$.I think answer is no.Theres a chance to be not constant.
Is my answers are true ?
Thanks

2. Jun 17, 2015

### ecastro

I think 'c' is a conceptual question. You don't need any form of equation to answer the question.

3. Jun 17, 2015

### RyanH42

4. Jun 17, 2015

### haruspex

Your reasoning for c) is good.
I can't comment on a) and b) because I'm not sure what the given equation for R is saying, but you appear happy with your answers to those.

5. Jun 17, 2015

### ecastro

Sorry about that. Here, to clear things up.

a.) Yes, this is correct. The velocity of the particle is constant if the first derivative of your position vector is constant, or if its second derivative is zero (both with respect to time).
c.) This is conceptual if you consider the question for general cases, that is, for all position vectors with a constant velocity. However, if the question is only for this problem in particular, you can calculate $\theta$.

6. Jun 18, 2015

### RyanH42

Ok,I get the idea.I can calculate the $cosθ$ Actually question b is asked for that I guessThanks for help.