Q.If the speed of light is $3.0 \times 10^8\ m\ s^{-1}$, calculate the distance covered by light in 2.00 ns.

The key idea is the speed-distance-time relation: distance = speed × time. The only catch is that time is given in nanoseconds, so you must convert to seconds first.

1. Convert 2.00 ns to seconds:

   $1\ \text{ns} = 10^{-9}\ \text{s}$, so $2.00\ \text{ns} = 2.00 \times 10^{-9}\ \text{s}$.

2. Apply the formula:

   $\text{distance} = (3.0 \times 10^8\ \text{m/s}) \times (2.00 \times 10^{-9}\ \text{s})$.

3. Multiply: …

The problem is a straightforward application of the Speed, Distance, Time relationship — one of the most fundamental ideas in physics. The trick here is not the formula itself, but handling the units carefully. Light travels at an enormous speed, and the time given is in nanoseconds (ns), a very small unit. If you plug in the numbers without converting, you'll get a wildly wrong answer.

Let’s break it down.

1. Recall the basic relation. The distance $d$ travelled by an object moving at constant speed $v$ in time $t$ is:

Here, $v = 3.0 \times 10^8\ \text{m/s}$ (the speed of light) and $t = 2.00\ \text{ns}$.

2. Convert nanoseconds to seconds. The prefix "nano" means $10^{-9}$. So:

Therefore:

3. Plug into the formula.

4. Multiply the numbers and the powers of ten separately. First, the coefficients: $3.0 \times 2.00 = 6.0$ …

Here are the common mistakes students make on this Speed-Distance-Time problem, and how to avoid each.

1. Forgetting to convert nanoseconds to seconds

The Mistake:

Plugging in $t = 2.00$ directly into $d = v \times t$, treating nanoseconds as seconds.

This gives a tiny, wrong answer:

$d = 3.0 \times 10^8 \times 2.00 = 6.0 \times 10^8\ \text{m}$ — which is incorrect because the time unit is wrong.

Why it happens:

Students see a number and forget that nano ($n$) means $10^{-9}$. The unit is ns, not s.

How to avoid:

Always write the unit conversion explicitly before substituting.

$1\ \text{ns} = 10^{-9}\ \text{s}$

So $2.00\ \text{ns} = 2.00 \times 10^{-9}\ \text{s}$

Then use:

2. Mishandling powers of ten (exponent arithmetic)

The Mistake:

Adding exponents incorrectly — e.g., writing $10^8 \times 10^{-9} = 10^{-1}$ (correct) but then misplacing the decimal in the final answer.

Why it happens:

Rushing through the multiplication of coefficients and exponents separately.

How to avoid:

Separate the work into two clear steps:

1. Multiply the coefficients: $3.0 \times 2.00 = 6.0$
2. Multiply the powers of ten: $10^8 \times 10^{-9} = 10^{8 + (-9)} = 10^{-1}$

Then combine:

3. Reporting the answer in wrong units

The Mistake:

Giving the answer in kilometres, centimetres, or leaving it without units — e.g., writing $0.6$ instead of $0.6\ \text{m}$.

Why it happens:

Not checking what the question asks for. The speed is in m/s, so distance naturally comes out in metres.

How to avoid:

Always write the unit after every numerical answer. If the question doesn’t specify a unit, use the one that follows from the given data — here, metres.

4. Confusing nanoseconds with microseconds or milliseconds

The Mistake:

Using $10^{-6}$ (micro) or $10^{-3}$ (milli) instead of $10^{-9}$.

Why it happens:

Mixing up metric prefixes — nano, micro, milli are easy to confuse under time pressure.

How to avoid:

Memorise the common prefixes with a mnemonic or a quick reference:

| Prefix | Symbol | Power | …