For a given initial projectile speed, you observe that the projectile has a certain range R at a launch angle A = 30 degrees. For what other launch angle will the projectile have the same range (assuming the same initial projectile speed)? Without the use of mathematical formula give a conceptual argument as to why this angle gives the same range (think about the tradeoffs between time-of-flight and horizontal component of velocity in your answer).

shoot straight at the target with a little up angle for drop or shoot almost straight up so it gets the target on the way down. There are two solutions for the equations that yield time for given height of target.

( Fire the ball or lob it :)

Vi = Vo sin T
u = Vo cos T

v = Vi - 9.8 t
v at top = 0
t = Vi/9.8 = (Vo/9.8)sin T
2 t = time in air (Vo/4.9) sin T

range = u (2 T)
= Vo cos T (Vo/4.9) sin T
= (Vo^2/4.9) cos T sin T

if T = 30
sin T = .5 and cos T = .866
but if T = 60
sin T = .866 and cos T = .5
same range

To understand why the range of a projectile remains the same for two different launch angles with the same initial projectile speed, we can consider the tradeoffs between the time of flight and the horizontal component of velocity.

The range of a projectile is determined by the product of the time of flight and the horizontal component of velocity. The time of flight is the duration it takes for the projectile to reach the ground, and the horizontal component of velocity determines how far the projectile travels horizontally during that time.

When the launch angle is increased from a smaller angle, such as 30 degrees, to a larger angle, the time of flight of the projectile decreases. This is because the vertical component of velocity increases, causing the projectile to reach the maximum height more quickly and subsequently spend less time in the air.

However, as the launch angle increases, the horizontal component of velocity also increases. This means that although the projectile spends less time in the air, it covers a greater horizontal distance during that time due to its higher horizontal velocity.

Therefore, when the launch angle is increased to a larger angle, the decreased time of flight is compensated by the increased horizontal component of velocity. As a result, the range of the projectile remains the same.

In conclusion, the range of a projectile stays constant for two different launch angles with the same initial projectile speed because the decrease in time of flight is balanced by the increase in horizontal component of velocity.

To understand why the projectile will have the same range for a different launch angle, let's consider the tradeoff between time-of-flight and the horizontal component of velocity.

When a projectile is launched at an angle, it has two components of motion: the vertical component and the horizontal component. The vertical component is affected by gravity, causing the projectile to follow a parabolic trajectory. The horizontal component determines the projectile's range.

Now, let's think about the tradeoff between time-of-flight and the horizontal component of velocity.

The time-of-flight is the total time the projectile spends in the air. It depends on the initial speed and the launch angle. At a launch angle of 30 degrees, the projectile stays in the air for a certain amount of time to reach its maximum height and then comes back down to the same height at a symmetrical point in its trajectory.

The horizontal component of velocity determines how far the projectile will travel horizontally before hitting the ground. As the angle of launch changes, the horizontal component of velocity will also change.

Considering this tradeoff, we can deduce that for a given initial launch speed and range, the time-of-flight and horizontal component of velocity must balance out. This means that for any angle, the time it takes for the projectile to reach its maximum height and come back down to the same height will be the same. Consequently, the horizontal distance covered by the projectile will also be the same.

Therefore, if the projectile has a certain range R at a launch angle of 30 degrees, it will have the same range for the launch angle that corresponds to the same time-of-flight. This is because the time-of-flight governs the range, regardless of the launch angle.

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