How Does Initial Bullet Speed Relate to Conservation of Energy and Momentum?

[hauthuong]

A bullet of mass 0.0021 kg is shot into a wooden block of mass 0.197 kg.

They rise to a final height of 0.546 m as shown. What was the initial speed (in m/s) of the bullet before it hit the block?

I figured out this one
I got another one
A 5.0 kg ball is attached to a vertical unstretched spring with k = 762 N/m and then released. What distance does the ball fall just as it momentarily comes to rest?
I use F=-kx with F=mg therefore mg=-kx from there I solve for x but I got the wrong answer. What am I wrong. Thank you

 

[Andrew Mason]

A 5.0 kg ball is attached to a vertical unstretched spring with k = 762 N/m and then released. What distance does the ball fall just as it momentarily comes to rest?
I use F=-kx with F=mg therefore mg=-kx from there I solve for x but I got the wrong answer. What am I wrong. Thank you

Think of it as a conversion of gravitational potential energy into spring potential energy and ball kinetic energy, as the ball falls.
$$mgx = \frac{1}{2}(kx^2 + mv^2)$$

Since maximum displacement occurs when v=0,
$$mgx = \frac{1}{2}kx^2$$

$$x = 2mg/k$$

AM

 

[hauthuong]

thank you
A 1.72 kg object is suspended from a spring with k = 19.1 N/m. The mass is pulled 0.305 m downward from its equilibrium position and allowed to oscillate. What is the maximum kinentic energy of the object in J?
1/2 kx^2=1/2 mv^2
and I got it wrong too

 

[Yegor]

Try to use 1/2 kx^2=1/2 mv^2+mgl, l - heigt
Because when the mass oscillates, it's potential energy changes

 

[Andrew Mason]

thank you
A 1.72 kg object is suspended from a spring with k = 19.1 N/m. The mass is pulled 0.305 m downward from its equilibrium position and allowed to oscillate. What is the maximum kinentic energy of the object in J?
1/2 kx^2=1/2 mv^2
and I got it wrong too

The condition for maximum kinetic energy is dK/dt = 0 (when the rate of change of kinetic energy = 0). Since $K = \frac{1}{2}mv^2 = \frac{1}{2}m(dx/dt)^2$ the maximum kinetic energy occurs when $d^2x/dt^2 = 0$ (ie. when a = 0).

The equation of motion is:
$$F = mg-Kx = ma$$ where x = the displacement from equilibrium

So when a=0
$$kx=mg$$
$$x=mg/k$$

Note that it is independent of the maximum amplitude. To find the speed when x=mg/k, use an energy approach:
$$U_g + KE + U_k = U_{ki}$$
$$mg(A-x) + \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$$


AM

 

[Yegor]

I'm afraid, that I didn't understand all correctly, but the question is "What is the maximum kinentic energy of the object in J?", isn't it?
I think You are right speaking about condition for maximum kinetic energy, however it depends on amplitude x.

1/2 kx^2=1/2 mv^2+mgx (conversion of energy)

1/2 mv^2=x(kx/2-mg)

 

[Andrew Mason]

I'm afraid, that I didn't understand all correctly, but the question is "What is the maximum kinentic energy of the object in J?", isn't it?
I think You are right speaking about condition for maximum kinetic energy, however it depends on amplitude x.

1/2 kx^2=1/2 mv^2+mgx (conversion of energy)

1/2 mv^2=x(kx/2-mg)

see my edited reply above.

AM

 

[Yegor]

But i have
mgx+(1/2)mv^2=(1/2)k(A-x)^2,
where A=(mg/k+x)
What do You think?

 

[Andrew Mason]

But i have
mgx+(1/2)mv^2=(1/2)k(A-x)^2,
where A=(mg/k+x)
What do You think?

A= initial amplitude.
Try:
$$mg(A-x) + \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$$

substituting x=mg/k:

$$\frac{1}{2}mv^2 = \frac{1}{2}kA^2 - mg(A - mg/k) - \frac{1}{2}m^2g^2/k$$

$$v^2 = KA^2/m - 2gA + mg^2/k$$
$$v = \sqrt{KA^2/m - 2gA + mg^2/k}$$

AM