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Let two similar vehicles starting at rest in a straight line (with a distance of $D$ between them) have time-varying accelerations in the same direction. If both of them have similar time varying accelerations, then can we say that the distance $D$ between them is constant? Moreover, if both of them have similar time varying accelerations, then are their velocities the same?

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There's a simple way to look at this that doesn't involve any maths.

Suppose the two cars are parked and are stationary, and you accelerate past them in your car. If you are accelerating forwards then from your perspective it looks as if the two cars are accelerating backwards (at the same rate). But the cars are at rest, so the distance between them can't be changing. This means that if two objects accelerate at exactly the same rate the distance between them will not change.

It's a simple extension of this to answer your question about velocities. Suppose now the two cars are travelling at the same velocity, and again you accelerate past them in your car. Once again it appears to you as if the two cars are accelerating backwards. But we know they are just travelling at the same speed, and this means that if two objects are accelerating at exactly the same rate their relative velocity doesn't change.

proving this mathematically is easy for us physics nerds, though possibly less so for non-physicists. Shout if you want me to show the maths and I'll add it to this answer.

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I assume that there is a silent assumption that all the time both cars are moving in the same direction. Otherwise, distance between them is not going to be constant. –  Wojciech Mar 28 at 9:53
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If the're moving at the same velocity they have to be moving in the same direction. –  John Rennie Mar 28 at 9:58
    
Ah, OK, I did use the words same speed rather than same velocity. Now corrected, thanks :-) –  John Rennie Mar 28 at 9:59
    
I just wanted to clarify that, because if people declare that they're not good at physics, they may not realize that velocity is a vector quantity, with all its consequences :) –  Wojciech Mar 28 at 10:02
    
Thanks a lot John and Wojciech for your replies. Yes, the assumption is that both the cars are moving in the same direction. –  Shaik Mar 28 at 10:54

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