Projectile Motion and Prime Axis

[kathmill]

The Problem
A projectile is thrown from a sloping hill with an initial speed of 20m/s directed perpendicular from the slope. If the incline of the slope is 32 degrees, how far from where it is thrown will the ball land?

I found these equations will result in a solution...
Want: R = square root of: (change in x)^2 + (change in y)^2 (pythagorean)
delta X= 1/2Axt^2
delta Y= Vyt + 1/2Ayt^2

tan(angle) = delta y/delta x

BUT...
where do I find Ax and Ay with the prime system?

 

[LowlyPion]

I think finding the solution through ordinary means in the x,y is probably a little easier than translating the axes.

If the initial velocity is perpendicular to the incline then you have your angle of launch and the component velocities readily enough.

If you treat then the incline as a function such that y = m*x where m is the slope, you can solve for the intersection of the trajectory and the slope.

 

[thomate1]

In order to solve this problem using the prime system, you will need to first understand the concept of prime axes. Prime axes are a coordinate system that is aligned with the initial velocity of the projectile. This means that the x-axis is parallel to the initial velocity and the y-axis is perpendicular to the initial velocity.

To find the values for Ax and Ay in this system, you can use trigonometric functions. Since the initial velocity is directed perpendicular to the slope, Ax will be equal to the initial velocity (20m/s) and Ay will be equal to 0.

Using the equations you have mentioned, you can now solve for the distance the projectile will travel using the prime axes coordinate system. Remember to use the values of Ax and Ay that you have determined.

R = √[(1/2Ax*t^2)^2 + (Vyt + 1/2Ay*t^2)^2]

In this case, t is the time the projectile is in the air, which can be calculated using the equation t = V*sin(angle)/g, where V is the initial velocity and g is the acceleration due to gravity.

I hope this helps you solve the problem and understand how to use the prime axes coordinate system in projectile motion calculations. Keep in mind that this system can be useful when the initial velocity is not directed along the traditional x and y axes.