To solve this problem, we need to determine the initial speed of the object using the given maximum height.

Given Information

1. Angle of projection ($\theta$) = $44^\circ$
2. Maximum height ($H$) = $4.5 \, \text{m}$
3. Gravitational acceleration ($g$) = $9.8 \, \text{m/s}^2$

Key Concept

The maximum height $H$ of a projectile launched at an angle $\theta$ with initial velocity $v_0$ can be calculated using the formula: $H = \frac{{(v_0 \sin \theta)^2}}{{2g}}$

Rearranging to solve for $v_0$: $v_0 = \sqrt{\frac{{2gH}}{{\sin^2 \theta}}}$

Substituting Values

1. $H = 4.5$ m
2. $g = 9.8$ m/s²
3. $\theta = 44^\circ$, so $\sin 44^\circ \approx 0.694$

Plugging in these values: $v_0 = \sqrt{\frac{{2 \times 9.8 \times 4.5}}{{(0.694)^2}}}$

Calculating step-by-step:

1. $2 \times 9.8 \times 4.5 = 88.2$
2. $(0.694)^2 \approx 0.481$
3. $\frac{88.2}{0.481} \approx 183.37$
4. $v_0 \approx \sqrt{183.37} \approx 13.54 \, \text{m/s}$

Conclusion

The closest answer from the options provided is 14 m/s.

Would you like further details or have any questions?

Here are five related questions for deeper understanding:

1. How would the initial speed change if the angle of projection increased?
2. What would be the effect on the maximum height if the launch angle was decreased?
3. How is the horizontal range of a projectile related to its initial speed?
4. How does gravity affect the maximum height of a projectile?
5. Can the same initial speed achieve different maximum heights? Why or why not?

Tip: In projectile motion, the vertical and horizontal components of velocity are treated separately, which simplifies solving for specific parameters like maximum height or range.