Find initial velocity of a projectile (v0) given displacement, time and acceleration?

goombachu

1. The problem statement, all variables and given/known data

I have a projectile that shoots straight up into the air (parallel to the y-axis). I need to calculate the initial velocity needed to reach a specific height. I am given the following:
- the y-displacement (s1) to reach
- the time (t) at which the projectile reaches the specified height
- the acceleration due to gravity (a = -9.8 m/s/s)

Additionally, after time (t), velocity should be zero, and the projectile should be at the specified height. In other words, it takes (t) seconds to reach the "apex."

2. Relevant equations

Nothing given, but I suspect:
s1 = s0 + (v0 * t) + (0.5 * a * t * t)

3. The attempt at a solution

Let's say:
- s0 = 0 meters
- s1 = 1 meters
- t = 0.25 secs
- a = -9.8 m/s/s
- v0 = ?

I thought I'd solve for v0 in the equation above. That gives:

v0 = s1 / t - (0.5 * a * t)
v0 = 1 / 0.25 - (0.5 * -9.8 * 0.25)
v0 = 4 - (-1.225)
=> v0 = 5.1225

That seems valid, but now let's say I need to update the velocity 30 times per second. This means that I am starting with an initial velocity of 5.1225 m/s, and each "tick" I need to decrement the velocity by some amount. After 0.25 seconds my velocity should tick down to zero.

I am guessing I would use (v1 = v0 + a * t) per "tick" for this, correct? Where v0 is always 5.1225 and (t) increments by (1/30) in each tick?

LowlyPion

You can use the V = Vo -a*t (treat a as negative) relationship at each 1/30 to give you V at each point leading up to 0. You can merely note in that regard that you will be decrementing by 9.8/30 = .32667m/s for each 1/30

goombachu

>> You can merely note in that regard that you will be decrementing by 9.8/30 = .32667m/s for each 1/30

That is what I thought as well, but:

0.25 secs * 30 ticks per second = ~8 ticks
0.32667 m/s * 8 ticks = 2.61336 meters

So my velocity doesn't tick down to zero in the 0.25 seconds I have to get there.

LowlyPion

That's because you didn't calculate the maximum height. You derived an equation for what the initial velocity would be if after 1/4 sec it got to height 1 meter.

To calculate initial Velocity if the max height is 1 m then you would choose V² = 2*(9.8)*1
That means V = 4.427m/s and the time to do that is x = 1/2*9.8*t² and the time then is .4518 s

goombachu

OK maybe I should rephrase the problem then to be clear. I need to velocity to hit zero at a predetermined height (1 meter in this example), and it needs to take exactly a predetermined amount of time (.25 seconds in this example) to get to that height. A negative force (which I thought could be gravity) must act on the projectile in order to launch it at a high speed and slow it down as it gets closer to the height.

Need to figure out:

a) What velocity to launch the projectile at in order to reach a given height at a given time
b) How to decrease the velocity in each 1/30 tick in order to reach velocity = 0 at a given time

The method you suggest looks like it doesn't get me to the predetermined height in the predetermined amount of time. At first glance it would seem that I might need to change the acceleration in order to do that?

LowlyPion

Looks like that's all that's left that you can change then.

On a planet far, far, away there is a boy playing with a ball ...

goombachu

Right. So now i have 2 unknowns then: initial velocity and acceleration.

I don't readily see any of the standard kinematic equations that could help me find both? The only one that doesn't involve both unknown terms is d = avg_vel * t, but that doesn't really give me an initial "launch" velocity does it?