Domain Analysis: Finding Valid Inputs for a Time-Dependent Velocity Function

Function Behavior with Physical Context

Which domain corresponds to the described function:

The function represents the velocity of a stone after being dropped from a great height as a function of time.

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Step-by-step written solution

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1

Understand the problem

Which domain corresponds to the described function:

The function represents the velocity of a stone after being dropped from a great height as a function of time.

2

Step-by-step solution

According to logic, the speed of the stone during a fall from a great height will increase as it falls with acceleration.

In other words, the speed of the stone increases, so the appropriate domain for this function is - always increasing.

3

Final Answer

Always increasing

Key Points to Remember

Essential concepts to master this topic
  • Physics Principle: Gravity causes constant downward acceleration for falling objects
  • Velocity Pattern: Speed increases continuously as v=gt v = gt where g = 9.8 m/s²
  • Domain Check: Time starts at t = 0 and velocity grows without bound ✓

Common Mistakes

Avoid these frequent errors
  • Confusing velocity with position or acceleration
    Don't think velocity decreases because the stone gets closer to ground = wrong physical understanding! Velocity is speed, not distance. Always remember that gravity accelerates objects downward, making them move faster over time.

Practice Quiz

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Is the function in the graph decreasing? yx

FAQ

Everything you need to know about this question

Why does the stone's velocity keep increasing instead of staying constant?

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Gravity acts as a constant force pulling the stone downward! Unlike moving at steady speed on flat ground, falling objects accelerate at 9.8 m/s² every second.

What if there was air resistance - would the function still be always increasing?

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Great question! With significant air resistance, velocity would eventually reach a terminal velocity and level off. But this problem assumes ideal conditions with negligible air resistance.

How is this different from throwing a stone upward?

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If thrown upward, velocity would first decrease (slowing down going up), reach zero at the peak, then increase while falling down. This problem specifies the stone is dropped, so it only falls.

Does the height matter for determining if velocity increases?

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The rate of increase stays the same regardless of height (9.8 m/s²), but dropping from greater height means more time to accelerate, reaching higher final speeds.

What does 'always increasing' mean mathematically?

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A function is always increasing when its output values get larger as the input increases. Here, as time passes (input), velocity gets faster (output grows).

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