Solving for time in 1D Kinematics

[Selroth]

Short Version:

It's been several years since I last practiced any mathematics or physics. I'm trying to get my mind as sharp as it was back then. I'm sure the solution is obvious, and just under my nose...

I remember:

P = P' + V't + (at^2)/2

Where P is position, P' is initial position, V' is initial Velocity, a is acceleration, and t is time.

Solve for t. All other variables are known.

I feel so embarrassed I don't remember how to do this.

Long Version (programmers may be interested):

Before I joined the military I had a good hobby of computer programming, and did well with mathematics and physics. Now, my mental capacity is at an all time low, and I thought picking my hobby back up again would do me good. The past few weeks have been great! But then I hit a brick wall, and 2 days of Googling and still no answer is starting to affect my enthusiasm. It's not like me to sign up to a physics forum and create an account just to ask one question, but perhaps it's not so bad to call upon the help of charitable others if you're thankful.

Anyway, my current project is simple, and nothing special. It's not meant to be. It'd not the product I expect to benefit from; it's the journey. So, it's really just a bouncing ball – I wanted to implement some collision detection ideas I've had. My main idea is instead of looking for collision by seeing if collidable objects are overlapping and correct them, calculate when they'll collide. Queue a collision event in a timeline using that time, and execute the collision reaction code once that time comes.

This is my latest build that's a good example: http://selroth.draconity.org/Projects/Active/White Sands/Javascript/20110930/

The vertical chart on the right is the timeline, and you can see the collision events crossing the cyan line at the moment the ball collides with the wall.

However, it's actually flawed/not fully functional in that aspect yet, despite initial impressions. Because the time of collision is always changing when there's an acceleration, it's difficult for the program to move the timeline to the exact time it needs to be at. Currently, it takes the time simply by dividing the distance between the wall and the ball by the ball's velocity.

I'm at a point where I need it to account for the ball's acceleration too – the event marks on the timeline shouldn't move around.

(current version can be found here: http://selroth.draconity.org/Projects/Active/White Sands/Javascript/Current/ which has better comments and cleaner code if you want to look at the sourcecode yourself [Chromers can Shift+Ctrl+J, Firefoxers can FireBug, IE or Safari'ers can get a better browser =P] )

Otherwise, this is the function I'm having trouble with:

	//Find the time of collision for either the top or bottom walls
	function findYCollision(){ return (yVel < 0) ? "top wall" : "bottom wall"; }	
	function findYCollisionTime(){
		var d = (findYCollision() == "top wall") ? yPos :  400 - yPos;
		var t = gameTime + (d / Math.abs(yVel));
		return (t > gameTime) ? t : gameTime;
	}

I can't figure out how to rewrite t = ______ to where it accounts for acceleration (would be known as yAcc in this case)

I am extremely grateful for any replies, and have marked this topic for email notifications! Thank you!

 

[Selroth]

I can't f'n believe this. Right when I posted my thread, for "similar topics" at the bottom of the page I find this: https://www.physicsforums.com/showthread.php?t=113313

That makes me want to scream and bang my head against the wall!

 

[Redbelly98]

Welcome to Physics Forums

That makes me want to scream and bang my head against the wall!

Ah, and welcome (back) to the world of physics

 

[Selroth]

Welcome to Physics Forums

Ah, and welcome (back) to the world of physics

Thanks! :)

 

[PJC01]

Dear fellow scientist,

Thank you for reaching out for help with your project and for sharing your journey back into mathematics and physics. It is completely normal to forget some concepts after a break, and it takes practice to regain your sharpness in these fields. I am happy to assist you in solving for time in 1D kinematics.

Firstly, let's review the equation you mentioned: P = P' + V't + (at^2)/2. This is a kinematic equation used to calculate the position of an object in motion, where P is the final position, P' is the initial position, V' is the initial velocity, a is the acceleration, and t is the time.

In order to solve for time, we need to rearrange the equation to isolate t on one side. We can start by subtracting P' from both sides to get P - P' = V't + (at^2)/2. Then, we can factor out t on the right side to get P - P' = t(V' + at/2). Finally, we can divide both sides by (V' + at/2) to get t = (P - P')/(V' + at/2). This is the solution for time in 1D kinematics.

Now, let's apply this to your specific problem. You mentioned that you are trying to find the time of collision for a bouncing ball. In this case, P is the final position of the ball when it collides with the wall, P' is the initial position of the ball, V' is the initial velocity of the ball, a is the acceleration of the ball, and t is the time of collision.

To solve for t, we need to know the values of P, P', V', and a. P and P' can be determined from the position of the ball and the wall, respectively. V' can be calculated using the initial velocity of the ball and the direction of motion (up or down). And finally, the acceleration can be calculated using the formula a = (Vf - Vi)/t, where Vf is the final velocity of the ball after the collision and Vi is the initial velocity of the ball.

I hope this helps you in solving for time in your project. Remember to keep practicing and don't be discouraged if you encounter challenges along the way. Science and mathematics require patience and perseverance. Best of luck to you!

Sincerely