
zerschlagen
to smash
To break or shatter something into many pieces with great force.
Der Einbrecher hat die Fensterscheibe mit einem Stein zerschlagen.
The burglar smashed the windowpane with a stone.
to smash / break up
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to smash
To break or shatter something into many pieces with great force.
Der Einbrecher hat die Fensterscheibe mit einem Stein zerschlagen.
The burglar smashed the windowpane with a stone.

to break up
To forcibly or permanently dissolve and destroy an organization, structure, or plan.
Die Polizei hat den Drogenring erfolgreich zerschlagen.
The police successfully broke up the drug ring.

to fall through
Used for plans or hopes: to not come about or to fail.
Unsere Urlaubspläne haben sich wegen der schlechten Wettervorhersage leider zerschlagen.
Unfortunately, our vacation plans fell through because of the bad weather forecast.
The prefix 'zer-' is inseparable and is never separated from the verb. It often expresses destruction or breaking apart, which can be seen in all meanings of 'zerschlagen' (to smash / break up).
Er hat aus Wut sein altes Handy zerschlagen.
Out of anger, he smashed his old phone.
In the physical meaning, 'zerschlagen' (to smash) is often used for fragile objects made of glass, ceramic, or porcelain. Typical objects are a bottle, a plate, or a windowpane.
Vorsicht, hier liegen Scherben, jemand hat eine Flasche zerschlagen.
Careful, there are shards here; someone smashed a bottle.
This meaning is metaphorical. Here, you do not 'zerschlagen' (smash) objects, but abstract structures. Typical objects are organizations such as a gang, a cartel, or a ring.
Der Polizei gelang es, den Schmugglerring zu zerschlagen.
The police managed to break up the smuggling ring.
For failed plans or hopes, the verb is used reflexively: 'sich zerschlagen' (to fall through). The subject is always the plan or the hope, not the person.
Mein Plan, am Wochenende wegzufahren, hat sich leider zerschlagen.
Unfortunately, my plan to go away for the weekend fell through.
to destroy
Both words can express that a plan can no longer be carried out. Zerstören describes generally how the plan is thwarted, while zerschlagen emphasizes actively and definitively putting an end to a plan. Use zerschlagen when the determined elimination of the plan is the focus.

Die neuen Vorschriften zerschlugen den gemeinsamen Reiseplan endgültig.
The new regulations definitively scuppered our travel plans.
Der Streit zerstörte den gemeinsamen Reiseplan.
The argument destroyed our travel plans.