At time t0 = 0 the mass happens to be at y0 = 4.05 cm and moving upward at velocity v0 = +4.12 m/s. (Mind the units!) Calculate the amplitude A of the oscillating mass. Answer in units of cm.

The amplitude of the oscillating mass is [tex]\(A = 4.05 \, \text{cm}\).[/tex]

To calculate the amplitude  A of the oscillating mass, we can use the equations of motion for simple harmonic motion (SHM). In SHM, the displacement [tex]\( y(t) \)[/tex] of the mass at time [tex]\( t \)[/tex] is given by:

[tex]\[ y(t) = A \sin(\omega t + \phi) \][/tex]

where:

- [tex]\( A \)[/tex]) is the amplitude,

- [tex]\( \omega \)[/tex] is the angular frequency, and

- [tex]\( \phi \)[/tex] is the phase angle.

Given that at [tex]\( t = t_0 = 0 \)[/tex], the mass is at [tex]\( y_0 = 4.05 \)[/tex] cm and moving upward at velocity [tex]\( v_0 = +4.12 \)[/tex] m/s, we can find the amplitude A

At [tex]\( t = 0 \),[/tex] we have:

[tex]\[ y(0) = A \sin(\phi) = y_0 = 4.05 \, \text{cm} \][/tex]

And also:

[tex]\[ v(0) = \omega A \cos(\phi) = v_0 = +4.12 \, \text{m/s} \][/tex]

To find [tex]\( A \),[/tex]A we'll use the fact that at [tex]\( t = 0 \)[/tex], the mass is at its maximum displacement, which means the velocity is zero at [tex]\( t = 0 \).[/tex] This gives us:

[tex]\[ \omega A \cos(\phi) = 0 \]\[ \cos(\phi) = 0 \][/tex]

Since [tex]\( \cos(\phi) = 0 \) when \( \phi = \frac{\pi}{2} \) or \( \phi = \frac{3\pi}{2} \), we'll consider \( \phi = \frac{\pi}{2} \).[/tex]

Now, from the equation [tex]\( A \sin(\phi) = y_0 \),[/tex] we can find [tex]\( A \):[/tex]

[tex]\[ A \sin\left(\frac{\pi}{2}\right) = 4.05 \]\[ A = 4.05 \, \text{cm} \][/tex]

So, the amplitude[tex]\( A \)[/tex] of the oscillating mass is [tex]\( 4.05 \, \text{cm} \).[/tex]

Two football players are pushing a 60 kg blocking sled across the field at a constant speed of 2.0 m/s. The coefficient of kinetic friction between the grass and the sled is 0.30. Once they stop pushing, how far will the sled slide before coming to rest?