The units of mass, length, and time being used are all halved.By what numerical factor does the measured value of the quantity increase by?

1. Break Down the Problem

To analyze how the halving of units affects the measured value of a physical quantity, we will consider quantities that depend on mass, length, and time. Common examples include force, energy, and speed. We will analyze each of these separately.

2. Relevant Concepts

• Force (F): $F = m \cdot a$ (Newton's second law), where $a$ is acceleration.
• Energy (E): For kinetic energy, $E_k = \frac{1}{2} m v^2$.
• Speed (v): $v = \frac{d}{t}$.

Here, when we halve the units:

• Mass is halved: $m' = \frac{m}{2}$
• Length is halved: $d' = \frac{d}{2}$
• Time is halved: $t' = \frac{t}{2}$

3. Analysis and Detail

1. Force:

   • Original: $F = m \cdot a$
     • Since acceleration $a$ is defined as $a = \frac{d}{t}$,
     • $a' = \frac{d'}{t'} = \frac{\frac{d}{2}}{\frac{t}{2}} = \frac{d}{t} = a$ (acceleration remains unchanged).
   • Therefore, $F' = m' \cdot a = \left(\frac{m}{2}\right) \cdot a = \frac{F}{2}$.

2. Kinetic Energy:

   • Original: $E_k = \frac{1}{2} m v^2$
   • Speed: $v = \frac{d}{t}$; on halving $d$ and $t$,
   • $v' = \frac{d'}{t'} = \frac{\frac{d}{2}}{\frac{t}{2}} = \frac{d}{t} = v$ (speed remains unchanged).
   • Therefore, $E_k' = \frac{1}{2} m' v'^2 = \frac{1}{2} \left(\frac{m}{2}\right) v^2 = \frac{E_k}{2}$.

3. Speed:

   • As mentioned above, $v' = \frac{d'}{t'} = \frac{\frac{d}{2}}{\frac{t}{2}} = \frac{d}{t} = v$ (speed remains unchanged).

4. Verify and Summarize

For both force and kinetic energy, when the units for mass, length, and time are halved, the measured values for both quantities decrease by a factor of 2. Speed remains unchanged.

Final Answer

The measured values of force and kinetic energy decrease by a factor of 2 when the units of mass, length, and time are all halved. Speed remains unchanged.