# How do you derive the correct projectile motion formula?

• gmy5011
In summary, the conversation discusses a problem involving a jaguar and a panther on an incline, and the goal is to determine the distance the jaguar jumps from the origin to intercept the panther. The conversation also touches upon the use of the general form of the equation s = s_0 + v_0(t) + 0.5a(t^2) in solving projectile motion problems and the importance of considering the height-dependence of velocity. It is also mentioned that experience can help in determining the most suitable approach to solve such problems.
gmy5011

## Homework Statement

A jaguar(A) leaps from the origin at a speed of v0 = 6 m/s and an angle β = 35° relative to the incline to try and intercept the panther(B) at point C. Determine the distance R that the jaguar jumps from the origin to point C. given the the angle of the incline is θ = 25°.

a = dv/dt

## The Attempt at a Solution

I know how to solve this problem by just looking up the constant acceleration formula and translating the velocity and R into cosines and sines. My question is, How do we know that we have to derive the equation s = s_0 +v_0(t) +.5a(t^2)? When I first tried it, and since it was trying to relate velocity and distance I thought I would use the derivation a = (dv/dx)(dx/dt) = v(dv/dx). Once I integrated it out I got a(x - x_0) = (v^2)/2 - (v_0^2)/2... why doesn't this work for solving the problem? How do you know from the beginning that you're going to have to use the general form I showed above when we are not given any info about time in the problem statement? Let me know if I need to clarify my question at all.

How do we know that we have to derive the equation s = s_0 +v_0(t) +.5a(t^2)?
You do not have to, but it is useful.

Once I integrated it out I got a(x - x_0) = (v^2)/2 - (v_0^2)/2... why doesn't this work for solving the problem?
How does the height-dependence of the velocity help?

How do you know from the beginning that you're going to have to use the general form I showed above when we are not given any info about time in the problem statement?
Experience. In doubt, use the general formula.

Thanks for the reply! Do we always have to use the general form for projectile motion since it is 2-D. I now kind of think we do, because we are going to need a way to relate the equations in both directions, so we are going to have to use a form with time because that's the variable that is the same in both directions, is that correct? Does that make sense?

There are always multiple methods to solve the problem, but that one is the easiest.

To derive the correct projectile motion formula, we need to start with the basic principles of motion and apply them to the specific situation given in the problem. In this case, we have a jaguar leaping from the origin at a given speed and angle, and we need to determine the distance it travels to intercept the panther at point C.

The first step in deriving the formula is to identify the forces acting on the jaguar. In this case, we have the force of gravity pulling the jaguar down and the force of the jaguar's initial velocity propelling it forward. These forces can be broken down into their components in the x and y directions.

Next, we need to apply Newton's second law, which states that the net force on an object is equal to its mass times its acceleration. In this case, the net force in the x direction is equal to the force of the jaguar's initial velocity, and in the y direction, the net force is equal to the force of gravity.

Using these principles, we can set up equations for the acceleration in the x and y directions. In the x direction, the acceleration is constant and equal to the jaguar's initial velocity. In the y direction, the acceleration changes due to the force of gravity.

By solving these equations for the position in the x and y directions, we can derive the formula for the distance traveled by the jaguar. This formula is s = s0 + v0t + 1/2at^2, where s0 is the initial position, v0 is the initial velocity, a is the acceleration, and t is the time.

In this problem, we are not given any information about time, but we can still use this formula by substituting in the known values for the initial velocity and acceleration. We can also use trigonometric functions to calculate the components of the initial velocity in the x and y directions.

In summary, to derive the correct projectile motion formula, we need to apply the principles of motion and forces to the specific situation given in the problem. This will lead us to the general formula for distance traveled, which we can then apply to the given values to solve for the distance R.

## 1. What is the projectile motion formula?

The projectile motion formula is a mathematical equation that describes the motion of an object that is projected into the air at an angle, such as a ball being thrown or a rocket being launched. It takes into account the initial velocity, angle of projection, and acceleration due to gravity to calculate the position of the object at any given time.

## 2. How is the projectile motion formula derived?

The projectile motion formula is derived using principles from both kinematics and calculus. It relies on the equations for displacement, velocity, and acceleration in the x and y directions, as well as the concept of vectors. By solving these equations simultaneously, we can arrive at the general formula for projectile motion.

## 3. What assumptions are made in deriving the projectile motion formula?

The projectile motion formula assumes that the object is moving in a vacuum, there is no air resistance, and the acceleration due to gravity is constant. It also assumes that the object is moving in a flat, horizontal plane and is not affected by any external forces besides gravity.

## 4. Are there any variations of the projectile motion formula?

Yes, there are variations of the projectile motion formula that take into account different scenarios, such as objects being launched from a height or on an incline. These variations may require additional equations or adjustments to the original formula, but the underlying principles remain the same.

## 5. How is the projectile motion formula used in real-world applications?

The projectile motion formula is used in various fields such as physics, engineering, and sports. It can be used to predict the trajectory and landing point of a projectile, which is important in designing and calibrating equipment such as missiles and sports equipment. It is also used in studying the motion of celestial bodies and understanding the behavior of objects in free fall.

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