why or how does an object emit radiation of different wavelength for a given temperature?
Google for "black body radiation", read what you find there, and if you have more specific questions or need help with some point, come back with a more focused questiojn - you'll get better answers faster that way.
The doubt is, how can a body emit radiations of different wavelength, for a given temperature ?
The thermal motions and other degrees of freedom occur at a wide variety of frequencies, and so energy is emitted at a variety of frequencies.
Temperature is a sort of macroscopic average across a very large number of microscopic particles doing very different things. Think of the steady buzz of conversation you hear in a crowded room when many different people are having many different conversations in many different tones of voices.
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