A baseball player swings and hits a pop fly straight up in the air to the catcher. The height of the baseball in meters $t$ seconds after it is hit is given by the quadratic function $h(t)=-4.9t^2+9.8t+1$. How long does it take for the baseball to reach its maximum height? What is the maximum height obtained by the baseball?

Step 1 ：The height of the baseball in meters $t$ seconds after it is hit is given by the quadratic function $h(t)=-4.9t^2+9.8t+1$.

Step 2 ：The maximum height of a parabolic function like this one is reached at its vertex. The $t$ value of the vertex of a parabola given in the form $f(t)=a{t}^2+bt+c$ is given by the formula $-\frac{b}{2a}$.

Step 3 ：In this case, $a=-4.9$ and $b=9.8$, so we can substitute these values into the formula to find the time at which the baseball reaches its maximum height.

Step 4 ：After finding the time, we can substitute it back into the function to find the maximum height.

Step 5 ：Substituting the values, we get $t_{max\mathrm{\_}height}=1.0$ and $max\mathrm{\_}height=5.9$.

Step 6 ：Final Answer: The baseball reaches its maximum height after $\boxed{{\displaystyle 1.0}}$ second. The maximum height obtained by the baseball is $\boxed{{\displaystyle 5.9}}$ meters.