Solving a Dynamics Problem - Projectile Question

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In summary, the conversation is about a projectile style question involving a man shooting another man who has climbed a pole. The task is to find an equation for the angle of projection needed for maximum range, given the initial velocity of the bullet and the distance from the shooter to the target. The person is struggling with the second part of the question but has the angle part sorted. They are looking for help in incorporating the tangent function into their equation. The final part of the conversation provides equations for horizontal and vertical velocity, displacement, and acceleration, along with a helpful reminder of trigonometric identities.
  • #1
Auron87
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Bit stuck on a projectile style question, at first glance I thought it would be pretty easy but I'm stuck now!

"A man is shooting another man who has climbed a pole h metres high, L metres away. The bullets are at speed v leaving the gun. Find an equation determining the angle of projection needed (alpha). Show that tan(alpha)=(v^2)gL for the maximum range."

Think I've got the angle part sorted, I have arccos(L/vt) for it anyway! But I'm really struggling getting the next part of the question, the nearest I've got is sin(2alpha) = (gL)/v^2 but can't really get tan into it! Any help would be much appreciated. Thanks.
 
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  • #2
Horizontal velocity, displacement and acceleration respectively:
[tex]v_x = v_0cos\alpha[/tex]
[tex]s_x = v_0tcos\alpha[/tex]
[tex]a_x = 0[/tex]

Vertical Velocity, displacement, acceleration respectively:
[tex]v_y = v_osin\alpha - gt [/tex]
[tex]s_y = v_0tsin\alpha - \frac{1}{2}gt^2 [/tex]
[tex]a_y = -g [/tex]

[tex]\therefore y = \frac{v_0xsin\alpha}{v_0cos\alpha} - \frac{1}{2}g[\frac{x}{v_0cos\alpha}]^2 [/tex]

Cancel down and re-arrange, and you should be able to get the answer. Just is case it is needed, [tex]sin2\alpha = 2sin\alpha.cos\alpha [/tex] and [tex]tan\alpha = \frac{sin\alpha}{cos\alpha} [/tex]
I trust that you can finish.
 
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1. What is a projectile?

A projectile is any object that is thrown or launched into the air and is subject to the forces of gravity and air resistance.

2. What is the equation for calculating the trajectory of a projectile?

The equation for calculating the trajectory of a projectile is: y = y0 + v0t + (1/2)at2, where y is the vertical displacement, y0 is the initial vertical position, v0 is the initial velocity, t is time, and a is acceleration due to gravity.

3. How do I solve a dynamics problem involving a projectile?

To solve a dynamics problem involving a projectile, you will need to break down the problem into smaller parts and apply the relevant equations, such as the kinematic equations, Newton's laws of motion, and the equations of motion for a projectile.

4. What factors affect the trajectory of a projectile?

The factors that affect the trajectory of a projectile include the initial velocity, angle of launch, air resistance, and the force of gravity.

5. How can I use a dynamics problem involving a projectile in real life?

Dynamics problems involving projectiles can be used in real life situations, such as calculating the trajectory of a baseball thrown by a pitcher, determining the landing spot of a golf ball hit by a golfer, or predicting the path of a missile launched by a military. They can also be used in engineering and physics to design and optimize the flight of projectiles in various applications.

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