Answer
a) $H=2.5m$
b) $H_1+H_2=1.98m$
Work Step by Step
a) As the block slides upward along the longer track, its KE decreases while PE increases. From the principle of energy conservation, we have $$\Delta KE = -\Delta PE$$ $$\frac{1}{2}m\Delta v^2=-mgH$$ $$\frac{1}{2} (v_f^2-v_0^2)=-gH (1)$$
The block has initial speed $v_0=7m/s$ and final speed $v_f=0$. From here, we can find $H$.
$$\frac{1}{2} (0-7^2)=-9.8H$$ $$H=2.5m$$
b) First, we find the speed at which the block leaves the track $v_f$, using equation (1): $$\frac{1}{2} (v_f^2-v_0^2)=-gH_1$$
We have $v_0=7m/s$ and $H_1=1.25m$, so $$\frac{1}{2} (v_f^2-7^2)=-9.8\times1.25=-12.25$$ $$v_f=4.95m/s$$
The block leaves the track at angle $\theta=50^o$, so its vertical speed is $v_y=4.95\sin50=3.79m/s$.
If we consider only vertical movement, the block reaches its highest point when $v_y=0$. So we can take $v_0=3.79m/s$, $v_f=0$ and use equation (1) to find $H_2$ $$\frac{1}{2} (0-3.79^2)=-9.8H_2$$ $$H_2=0.73m$$
So, $H_1+H_2=1.98m$
