This problem involves projectile motion. We need to find the initial speed ($u$) given the launch angle ($\alpha$) and the speed ($v$) at a specific angle ($\theta$) with the horizontal.
Projectile Motion Analysis
Given parameters:
- Launch angle, $\alpha = 60^\circ$
- Speed at a certain point, $v = 20$ m/s
- Angle of motion at that point, $\theta = 45^\circ$
- We need to find the initial speed, $u$.
Calculating Initial Speed
For projectile motion, a relationship between the initial speed ($u$), the speed ($v$) at a later point, the launch angle ($\alpha$), and the angle of motion ($\theta$) at that later point can be expressed. A relevant formula often used or derived for this scenario is:
$ u = v \frac{\tan \alpha}{\tan \theta} $
Substitute the given values into the formula:
$ u = 20 \, \text{m/s} \times \frac{\tan 60^\circ}{\tan 45^\circ} $
We know that $\tan 60^\circ = \sqrt{3}$ and $\tan 45^\circ = 1$. Plugging these values in:
$ u = 20 \times \frac{\sqrt{3}}{1} $
$ u = 20\sqrt{3} \, \text{m/s} $
Final Answer
The initial speed of the projectile is $20\sqrt{3}$ m/s.
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