Projectile motion maximum range

In summary, for a plane inclined at an angle A to the horizontal and a particle projected up the plane with a velocity u at an angle B to the plane, the range is a maximum when B = 1/2[(π/2) - A]. This can be proven by taking into consideration the x and y components of the projectile's motion and finding the angle that results in the largest sum of the two components. At 45 degrees, the sine and cosine values are equal, resulting in the maximum range.
  • #1
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A plane is inclined at an angle A to the horizontal. A particle is projected up the plane with a velocity u at an angle B to the plane. the plane of projection is vertical and contains the the line of greatest slope

Prove that the range is a maximum when [tex]\displaystyle{B=\frac{1}{2}[(\frac{\pi}{2})-A]}[/tex]

I found the time of flight to be [tex]\displaystyle{\frac{2uSinB}{gCosA}}[/tex] and then tried to find [tex]S_{x}[/tex] at this time. It was pretty long and I am not sure how to prove the range bit. Is it something to do with the maximum value of Sine being 1?

Thank you
 
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  • #2
I think you're on the right track with the values of sine. Since you can break up the motion of a projectile into x and y components by taking the sine and cosine of the same angle, the maximum range will be when the sum of both x and y components are their largest. The larger the y value, the more flight time there is; the larger the x value is, the more range it gets per unit time. For example, at 90 degrees to the horizontal: sine is 1 but cosine is zero, so the sum of sin(90) and cos(90) is 1 (shooting straight up, all y component, no x component, not maximum range). At 85 degrees to the horizontal: sin(85) = .996 and cos(85) = .087, sum is 1.083, small x componenet, large y componenet, not maximum range. At 45 degrees: sin(45) is .707 and cos(45) is .707, and their sum is 1.414 (maximum range).

This is just a guess/shot in the dark but I hope it helps.
 

1. What is projectile motion and how is it defined?

Projectile motion is the motion of an object through the air or space under the influence of gravity. It is defined as the path followed by a projectile, which is any object that is thrown, shot, or launched into the air.

2. What is the maximum range in projectile motion?

The maximum range in projectile motion is the farthest horizontal distance that a projectile can travel before hitting the ground. It occurs when the projectile is launched at an angle of 45 degrees with respect to the horizontal.

3. How is the maximum range affected by the initial velocity and angle of launch?

The maximum range is directly proportional to the initial velocity and the sine of the launch angle. This means that the higher the initial velocity and the smaller the launch angle, the greater the maximum range will be.

4. Can the maximum range be affected by air resistance?

Yes, air resistance can affect the maximum range in projectile motion. When air resistance is present, the projectile experiences a force in the opposite direction of its motion, causing it to slow down and decrease its maximum range.

5. What are some real-life applications of projectile motion and its maximum range?

Projectile motion and its maximum range have various real-life applications, including sports such as basketball, baseball, and golf, where players need to calculate the ideal angle and velocity for their shots. It is also used in military operations for estimating the range of projectiles and in engineering for designing and testing projectiles and launchers.

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