Answer
a) $k=1000N/m$
b) $\mu_k=0.34$
Work Step by Step
The block starts from $v_0=0$ to $v=5m/s$ in $t=0.5s$. The acceleration is $$a=\frac{v-v_0}{t}=10m/s^2$$
The spring is pulled with force $P$, which is opposed by restoring force $F_x$. The spring will stop stretching when $P=F_x=kx$, so
- First, the block accelerates: $P_1=kx_1=0.2k$
- Next, the block moves at a constant speed: $P_2=kx_2=0.05k$
The block is pulled with force $P$, which is opposed by kinetic frictional force $f_k$.
- First, the block accelerates at $a=10m/s^2$: $$P_1-f_k=ma$$ $$0.2k-\mu_kmg=ma$$ $$0.2k-(15kg)(9.8m/s^2)\mu_k=(15kg)(10m/s^2)$$ $$0.2k-147\mu_k=150 (1)$$
- Next, the block moves at a constant speed: $$P_2-f_k=0$$ $$0.05k-\mu_kmg=0$$ $$0.05k-147\mu_k=0 (2)$$
Solve (1) and (2), we get $k=1000N/m$ and $\mu_k=0.34$
