Advanced Math/more 2 dimensional motion
This is a question from Mechanics 1. I'm self studying so have no teacher to ask. Your previous response was so clear that I'd like help with this one. I have completed all other questions in the chapter but this one has me totally foxed, it's so different from the examples given and the rest on the questions.
An object is moving in a plane. At time t =0, it is at the origin O and moving with velocity u. After 2 seconds it is at A where OA(vector)= -2i -4j. After a further 3 second it is at B where AB(vector) is 10i - 40j.
Show that this is consistent with constant acceleration a, Find a and u.
I expected to have to use "u" somewhere in the journey from O to A an use it to find the velocity and acceleration at A. Then I expected to use the velocity just found to find the velocity at B using a different acceleration and then compare the accelerations to show that they were the same. I've tried this many times but end up with huge equations. It can't be right.
The book answer is a= (26i -68j)/15, u = (-41i+38j)/15
So, the task here is to show that the motion of the object is consistent with that of constant acceleration. We therefore need to find a and u such that the assumption of constant acceleration is satisfied.
As usual, we make use of the equations of motion with constant acceleration between any two chosen intervals. In the first 2 seconds;
ΔR = ut + Żat▓
-2i - 4j = 2u + 2a
a = -i - 2j - u
Considering the combined 5 seconds, again
ΔR = ut + Żat▓
(-2i - 4j) + (10i - 40j) = 5u + 12.5a
8i - 44j = 5u + 12.5a
12.5a = 8i - 44j - 5u
This is now just an issue of solving the equations simultaneously and finding a and u, the solutions to which i'm sure will agree with that from the book as you've provided.
To confirm the constant acceleration consistency you need to then use the equation(s) of motion to check the positions, velocities and/or times in the available intervals OA, AB and OB.
Note that using the interval AB in the initial solution would have required finding some intermediate velocity, the need for which we have avoided by considering interval OB and thereby saving some time and effort.
Its admirable that you're making the effort to educate yourself with these things and you can always get back to me.