There are lots of other things we can investigate with projectile motion. So, there we have it, we can now create the equation of the envelope of curves created by projectile motion for any given initial velocity! If I was to take an initial velocity of 2 then I would have the following:Īnd an initial velocity of 4 would generate the following graph: This is the envelope of projectile motion when I take the following projectiles in parametric form and vary theta from 0 to pi: For example, if I launch a projectile with a velocity of 1, and taking g = 9.81, I get the following equation: Let’s look at some of the envelopes it will create. We then have the difficulty of simplifying the second denominator, but luckily we have a trig equation to help:Īnd we have our equation for the envelope of projectile motion! As we can see it is itself a quadratic equation. We can then substitute this into the equation for F(x,y,theta)=0 to eliminate theta: We can then rearrange this to get x in terms of theta: Therefore we can rearrange our equation for y to give:Īnd in order to help find the partial differential of F we can write: This means that we differentiate as usual with regards to theta, but treat x and y like constants. The second of these equations means the partial derivative of F with respect to theta. Next, we use the fact that the envelope of a curve is given by the points which satisfy the following 2 equations:į(x,y,theta)=0 simply means we have rearranged an equation so that we have 3 variables on one side and have made this equal to 0. Here v is the initial velocity which we will keep constant, theta is the angle of launch which we will vary, and g is the gravitational constant which we will take as 9.81.įirst let’s rearrange these equations to eliminate the parameter t. Let’s start with the equations for projectile motion, usually given in parametric form: The black dotted line is then called the envelope of all these lines, and is the boundary line formed when I plot quadratics for every possible angle between 0 and pi.įinding the equation of an envelope for projectile motion In the graph above I have changed the launch angle to generate different quadratics. If you are a teacher then please also visit my new site: for over 2000+ pdf pages of resources for teaching IB maths!įor any given launch angle and for a fixed initial velocity we will get projectile motion.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |