A misconception I have about impulse formula interpretation

[elElegido]

The misconception came up from the following problem: "A 0.50-kg cart (#1) is pulled with a 1.0-N force for 1 second; another 0.50 kg cart (#2) is pulled with a 2.0 N-force for 0.50 seconds. Which cart (#1 or #2) has the greatest acceleration? "

I know the answer is the following (I looked it up) a=f/m --> 1N/0.5kg= 2m/s2 and 2N/0.5kg = 4m/s2, so the 2n carts acceleration is greater but for me it goes against common sense. What common sense tells me is that if you apply 2N for 0.5s causes the same acceleration as applying 1N for 1s if the object has in both cases has the same mass: 2N*0.5s = 1N*s average force and 1N*1s = 1N*s average force, this should cause the same acceleration for both, not acceleration #2cart >#1cart.

What's the point of saying we apply the for during let's say 1s, 5000000s, or in 0.0000000001s if the time doesn't really affect the acceleration? What's the point of impulse as force*time? What I see is acceleration doesn't care about how much time you apply a force so it goes against the concept of impulse itself, doesn't it?

Thanks in advance for your help. Have a great day.

[Moderator's note: moved from a technical forum.]

 

[PeroK]

The change in velocity is the same. But, as the change takes half the time in the second case, the acceleration must be greater. In other words, in terms of well-defined physics:
$$a = \frac{\Delta v}{\Delta t} \ \text{and} \ a \ne \Delta v$$

 

[Mister T]

What I see is acceleration doesn't care about how much time you apply a force so it goes against the concept of impulse itself, doesn't it?

No. The two concepts are completely compatible. It may be that one or the other is more useful in a particular scenario.