First, let's find the initial momentum of the steel ball and the large mass. The momentum of the steel ball is $p_{b, initial} = m_bv_b = (5 \times 10^{-3}\,\text{kg})(250\,\text{m/s}) = 1.25\,\text{kg m/s}$. Since the large mass is initially at rest, its initial momentum is zero.
After the collision, the steel ball and the large mass will have the same final momentum since no kinetic energy is lost. Let $v_{b, final}$ and $v_{M, final}$ be the final velocities of the steel ball and the large mass, respectively. The conservation of momentum equation is:
$$p_{b, initial} + p_{M, initial} = p_{b, final} + p_{M, final}$$
Since the initial momentum of the large mass is zero, the equation simplifies to:
$$1.25\,\text{kg m/s} = m_bv_{b, final} + Mv_{M, final}$$
Now, let's use the conservation of energy to find the relationship between the final velocities of the steel ball and the large mass. Half of the sum of the initial and final kinetic energies of the steel ball is stored as potential energy in the spring when it is compressed:
$$\frac{1}{2}m_b(v_{b, initial}^2 + v_{b, final}^2) = \frac{1}{2}k d^2$$
Since we are given that the steel ball's kinetic energy does not change, $v_{b, initial} = v_{b, final}$. Substituting this into the equation:
$$\frac{1}{2}(5 \times 10^{-3}\,\text{kg})(250\,\text{m/s})^2 = \frac{1}{2}(53\,\text{N/m}) d^2$$
Solving for $d^2$:
$$d^2 = \frac{(5 \times 10^{-3}\,\text{kg})(250\,\text{m/s})^2}{53\,\text{N/m}}$$
$$d^2 = 0.147\,\text{m}^2$$
Therefore, the distance $d = \sqrt{0.147\,\text{m}^2} = 0.383\,\text{m}$ (to three significant figures). The spring is compressed by 0.383 meters when the large mass comes briefly to rest.
A steel ball of mass m=5 g is moving at a speed of 250 m/s toward a large mass M=2.2 kg which is connected to a relaxed spring, as shown below. When the ball reaches the large mass it collides and bounces off without losing any kinetic energy. The large mass is connected to a spring of spring constant k=53 N/m and after the collision the large mass moves to the left, compressing the spring. The mass slides on a frictional surface with kinetic coefficient of friction 0.80. Find the distance d by which the spring is compressed when the large mass comes briefly to rest. d=?
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