Hello there, I'm a physics enthusiast with a keen interest in the dynamics of projectile motion. When it comes to the question of what launch angle gives the longest range, we delve into the realm of physics known as projectile motion, which is a type of motion experienced by an object that is launched near the Earth's surface and moves under the influence of gravity.

In the simplest terms, the range \( R \) of a projectile launched with an initial velocity \( v_0 \) at an angle \( \theta \) above the horizontal can be calculated using the formula:

\[ R = \frac{{v_0^2}}{{g}} \sin(2\theta) \]

where \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \) on Earth.

The sine function, as mentioned in the provided reference, reaches its maximum value of 1 when its argument is \( \frac{\pi}{2} \) radians, or 90 degrees. However, this is a slight misunderstanding of the context. The sine function is indeed maximized at 90 degrees, but in the context of projectile motion, we are interested in the angle at which the sine of twice that angle is maximized, because the range equation involves \( \sin(2\theta) \), not \( \sin(\theta) \).

To find the maximum range, we need to find the angle \( \theta \) that maximizes \( \sin(2\theta) \). Since the sine function is periodic and has a maximum value of 1, the maximum value of \( \sin(2\theta) \) will occur when \( 2\theta = \frac{\pi}{2} \), or \( \theta = \frac{\pi}{4} \) radians, which is equivalent to 45 degrees.

So, a projectile launched at an angle of 45 degrees will have the greatest range, assuming no air resistance and that the only force acting on the projectile after it is launched is gravity.

This conclusion is based on the principles of physics and the mathematical optimization of the range equation. It's important to note that in real-world scenarios, factors such as air resistance, the height from which the projectile is launched, and the initial speed can all affect the actual range achieved.

Now, let's move on to the translation of the answer into Chinese.

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The sine function reaches its largest output value, 1, with an input angle of 90 degrees, so we can see that for the longest-range punts 2-- = 90 degrees and, therefore, -- = 45 degrees. A projectile, in other words, travels the farthest when it is launched at an angle of 45 degrees.Nov 9, 2010read more >>