Answer
(a) The greatest speed the mass reaches is 3.03 m/s when the spring has reached its normal length. This occurs after the block moves a distance of 9.59 cm
(b) The greatest acceleration of the mass is $95.9~m/s^2$ and this occurs when the mass is initially released from rest.
Work Step by Step
(a) We can find the distance x that the spring is compressed.
$\frac{1}{2}kx^2 = 11.5~J$
$x^2 = \frac{23~J}{2500~N/m}$
$x = \sqrt{\frac{23~J}{2500~N/m}}$
$x = 0.0959~m$
We can find the greatest speed the mass reaches.
$\frac{1}{2}mv^2 = 11.5~J$
$v^2 = \frac{23~J}{2.50~kg}$
$v = \sqrt{\frac{23~J}{2.50~kg}}$
$v = 3.03~m/s$
The greatest speed the mass reaches is 3.03 m/s when the spring has reached its normal length. This occurs after the block moves a distance of 9.59 cm
(b) The greatest acceleration occurs when the spring pushes with its greatest force. This occurs when the block is released from rest.
$ma = kx$
$a = \frac{kx}{m} = \frac{(2500~N/m)(0.0959~m)}{2.50~kg}$
$a = 95.9~m/s^2$
The greatest acceleration of the mass is $95.9~m/s^2$ and this occurs when the mass is initially released from rest.
