Calculating Time for Block to Reach Max. Amp. After Spring Collision

[Momentum09]

Homework Statement

A block of mass 0.28kg is attached to a spring of spring constant 12N/m on a frictionless track. The block moves in simple harmonic motion with amplitude 0.2m. While passing through the equilibrium point from left to right, the block is struck by a bullet, which stops inside the block. The velocity of the bullet immediately before it strikes the block is 53m/s and the mass of the bullet is 4.4g.
How long will it take for the block to reach maximum amplitude [which I calculated to be 0.3246) after the collision?

Homework Equations

M1V1 + M2V2 = (M1+M2)Vf
x = Bsin(wt+alpha)
I found Vf = 1.309 m/s

The Attempt at a Solution

I think the answer will have to solved using the equation, x = Bsin (wt + alpha), but I don't know what numbers to plug in for those. Any help would be greatly appreciated!

 

[Momentum09]

I got the answer. :)

 

[m2003]

I would first clarify the question by asking for more information such as the initial position and velocity of the block before the collision. Without this information, it is impossible to accurately calculate the time it takes for the block to reach maximum amplitude after the collision.

However, assuming that the initial position and velocity of the block are both zero, we can use the conservation of momentum equation (M1V1 + M2V2 = (M1+M2)Vf) to find the final velocity of the block after the collision. Plugging in the values given, we get:

(0.0044 kg)(53 m/s) + (0.28 kg)(0 m/s) = (0.0044 kg + 0.28 kg)Vf
Vf = 1.309 m/s

Next, we can use the equation for simple harmonic motion (x = Bsin(wt + alpha)) to find the time it takes for the block to reach maximum amplitude. However, we need to know the value of the angular frequency (w) and the initial phase (alpha) in order to solve for time. These values can be calculated using the given information and the equation for angular frequency (w = sqrt(k/m)).

w = sqrt(12 N/m / 0.28 kg) = 5.305 rad/s
alpha = arctan(-Vf/wB) = arctan(-1.309 m/s / (5.305 rad/s)(0.2 m)) = -0.462 radians

Plugging these values into the equation for simple harmonic motion, we get:

0.3246 m = (0.2 m)sin(5.305 rad/s * t - 0.462 rad)
Solving for t, we get t = 0.057 seconds.

Therefore, it will take approximately 0.057 seconds for the block to reach maximum amplitude after the collision. However, as mentioned before, this answer may not be accurate without knowing the initial position and velocity of the block. It is important to always provide all necessary information when solving physics problems to ensure accurate and meaningful results.