Bullet acceleration into moist clay

[thehitchcocks]

The problem:
A bullet is moving at a speed of 367 m/s when it embeds into a lump of moist clay. The bullet penetrates for a distance of 0.0621m. Determine the acceleration of the bullet while moving into the clay.

I can get the correct answer by using V_f² = V_i² + 2ad (result: -1.08*10^-6 m/s²).

However, I'm trying to understand why an approach using the average velocity doesn't work (that's what I tried first). Can you help me understand what's wrong with the following approach? :

V_avg = (V_i + V_f)/2 = 183.5 m/s

t = d/v
t = 0.0621m / (183.5 m/s) = 0.00034s

d = V_it + 1/2at²
0.0621m = 183.5 m/s * 0.00034s + 1/2a(0.00034s)²
a = -5017.3 m/s²

(you can double-check my math, but I don't believe that's the issue).

So, my specific question: this approach clearly doesn't yield the correct answer - there's something wrong with my thinking here. What specifically is wrong with it?

 

[RUber]

Your math was wrong.

The problem:
A bullet is moving at a speed of 367 m/s when it embeds into a lump of moist clay. The bullet penetrates for a distance of 0.0621m. Determine the acceleration of the bullet while moving into the clay.

I can get the correct answer by using V_f² = V_i² + 2ad (result: -1.08*10^-6 m/s²).

However, I'm trying to understand why an approach using the average velocity doesn't work (that's what I tried first). Can you help me understand what's wrong with the following approach? :

V_avg = (V_i + V_f)/2 = 183.5 m/s

t = d/v
t = 0.0621m / (183.5 m/s) = 0.00034s

d = V_it + 1/2at²
0.0621m = 183.5 m/s * 0.00034s + 1/2a(0.00034s)²
a = -5017.3 m/s²

(you can double-check my math, but I don't believe that's the issue).

So, my specific question: this approach clearly doesn't yield the correct answer - there's something wrong with my thinking here. What specifically is wrong with it?

You use Vavg where you should use Vi.
Also, you have a negative exponent on what you call the correct answer.

 

[PeroK]

You plugged in average velocity instead of initial velocity.

If you don't mind my saying, I don't see any maths in your post: just lots of numbers. Part of the reason you made this mistake is that one number looks very like another and as soon as you plug in the numbers, you lose sight of the maths and physics.

 

[Buzz Bloom]

0.0621m = 183.5 m/s * 0.00034s + 1/2a(0.00034s)2

Hi thehitchcocks:

The term 183.5 m/s is not V_i. It is 1/2 V_i.

Regards,
Buzz

​

 

[thehitchcocks]

Thanks for all the replies! Super helpful. We understand now - we got ourselves wrapped up mistakenly in average velocity, when all we needed was initial velocity. Using initial velocity yields the correct answer.

@RUber: Yes, the negative exponent was a typo - sorry about that, our first post ever here, and I didn't catch the error amidst all the markup :( Thanks also for coming back and updating your original response, re: the relationship between our answer and the correct answer.