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## Homework Statement

So I've been stuck on this exercise for a few hours now, maybe you can help out:

3 balls meet at the same height h

_{m}.

Ball 1 is accelerated straight up into the air for 1s from height 0.

When the acceleration stops, ball 2 is launched straight up from height 0 with v

_{2}= 29.358 m/s.

A short time later ball 3 is dropped from height h

_{c}=60m.

Each ball has a mass of 1g. The height h

_{m}is the maximum height that ball 1 reaches on its flight path.

Friction can be neglected.

1) calculate h

_{m}and the acceleration a of ball 1.

2) what is the time between the moment, that ball 2 is launched and the moment ball 3 is dropped from the ceiling h

_{c}=60m?

## Homework Equations

Kinematic equations for constant acceleration

## The Attempt at a Solution

I defined:

t

_{0}as the time that the acceleration of ball 1 stops and ball 2 is launched with v

_{2}= 29.358 m/s

t

_{1}as the time that ball 3 is dropped from the ceiling = ?

t

_{2}as the time at which all balls are at height h

_{m}= ?

h

_{m}as the height at which all balls meet = ?

h

_{1}as the height at which the acceleration of ball 1 stops = ?

h

_{c}as the height of the ceiling = 60m

v

_{1}as the velocity of ball 1 when the acceleration stops = ?

v

_{2}as the initial velocity of ball 2 = 29.358 m/s

Then I tried to come up with kinematic equations for all 3 balls:

- Ball 1:

eq1: h

_{m}= h

_{1}+ v

_{1}*t

_{2}- 0.5*g*t

_{2}

^{2}

eq1.1: v

_{1}

^{2}= 2*g*(h

_{m}-h

_{1})

- Ball 2:

eq2: h

_{m}= v

_{2}*t

_{2}- 0.5*g*t

_{2}

^{2}

- Ball 3:

eq3: h

_{m}= h

_{c}- 0.5*g*(t

_{2}- t

_{1})

^{2}

I wrote the height of ball 1 as a function of time:

h(t) = -0.5*g*t

^{2}+ v

_{1}*t +h

_{1}

The height at which the balls meet is the max point of the function above so I set the first derivative = 0 and solved for t:

h'(t) = -g*t + v

_{1}= 0

So t = v

_{1}/g which should be t

_{2}

I tried out plugging that into equation 1 and 2 and I also tried to set eq1 = eq2 to solve for something, but from that point I just went in circles and I just can't figure out how to proceed. Maybe I'm blind to something obvious because I've been trying for so long to solve this.

I would really appreciate any help or hints on how to go on from there.

Thanks in advance for your time.

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