Deriving Projectile Motion Equations from Initial Conditions

In summary, the conversation is about deriving algebraic expressions for the range and total time-of-flight of a projectile launched with initial speed v, from a height h, and at an angle Q, above the horizontal. The equations for range and time-of-flight have been given, but the method of derivation is unknown. The first step in deriving the equations is to find the height as a function of time and set it to zero to find the time-of-flight.
  • #1
charris316
2
0

Homework Statement


Derive algebraic expressions for the range and total time-of-flight of a projectile launched with initial speed v, from a height h, and at an angle Q, above the horizontal. We were given the final equations but I am unsure of how to derive them.


Homework Equations


range=vt-.5gt^2, where g is the acceleration due to gravity and t is the time in seconds.

time= (vsinQ+((vsinQ)^2+2gh)^.5)/g


The Attempt at a Solution


Considering the ball, after it has been launched:
Fx=0
Vox=vcosQ
Fy=mg
Voy=vsinQ

?
 
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  • #2
That equation for range is wrong.

The first step is to find an the height as a function of time.
Set this to zero to find the time-of-flight.
 

1. What is the purpose of deriving a motion equation?

The purpose of deriving a motion equation is to describe the mathematical relationship between an object's position, velocity, and acceleration over time. It allows us to predict an object's motion and understand how it will change in different situations.

2. How do you derive a motion equation?

To derive a motion equation, you need to start with a set of motion data, such as an object's initial position, velocity, and acceleration. Then, you use the equations of motion and calculus methods to manipulate these variables and eliminate any unknowns until you are left with a single equation that represents the object's motion.

3. What are the equations of motion used in deriving a motion equation?

The three equations of motion are:
- x = x0 + v0t + (1/2)at2
- v = v0 + at
- v2 = v02 + 2a(x-x0)
where x is the position, v is the velocity, a is the acceleration, x0 and v0 are the initial position and velocity, and t is time.

4. What are some real-life applications of deriving motion equations?

Deriving motion equations has many practical applications, such as in engineering, physics, and even sports. It can be used to predict the trajectory of a projectile, design roller coasters, and analyze the motion of vehicles. It is also used in sports science to improve athletic performance and prevent injuries.

5. Are there any limitations to deriving motion equations?

While deriving motion equations can accurately predict an object's motion in ideal conditions, it may not account for real-life factors such as air resistance, friction, and human error. Additionally, it assumes that the object's acceleration remains constant, which may not always be the case. Therefore, it is important to use these equations with caution and consider other factors that may affect an object's motion.

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