Question

Event A occurs at space-time coordinates $(300\mathrm{m},2\mathrm{\mu s})$.

a. Event B occurs at space time coordinates $(1200\mathrm{m},6\mathrm{\mu s})$. Could A possibly be the cause of B? Explain.

b. Event C occurs at space time coordinates $(2400\mathrm{m},8\mathrm{\mu s})$. Could A possibly be the cause of C? Explain

Short answer

a. yes

b. no

Step-by-step solution

Part (a) Step 1: Given information

We have given, Space-time coordinates of A

$(300\mathrm{m},2\mathrm{\mu s})$

Space time coordinates of B

$(1200\mathrm{m},6\mathrm{\mu s})$

we have to find the is A possibly be the cause of B.

Simplify

The distance between the A and B

$$\begin{gathered} ∆\mathrm{x}={\mathrm{x}}_{\mathrm{B}}-{\mathrm{x}}_{\mathrm{A}} \\ ∆\mathrm{x}=1200\mathrm{m}-300\mathrm{m} \\ ∆\mathrm{x}=900\mathrm{m} \end{gathered}$$

The time taken from A to B

$$\begin{gathered} ∆\mathrm{t}={\mathrm{t}}_{\mathrm{A}}-{\mathrm{t}}_{\mathrm{B}} \\ ∆\mathrm{t}=6\mathrm{\mu s}-2\mathrm{\mu s} \\ ∆\mathrm{t}=4\mathrm{\mu s} \end{gathered}$$

then velocity will be

$$\begin{gathered} \mathrm{V}=\frac{∆\mathrm{x}}{∆\mathrm{t}}=\frac{900\mathrm{m}}{4\mathrm{\mu s}} \\ \mathrm{V}=225\times {10}^6\mathrm{m}/\mathrm{s} \\ \mathrm{V}<<\mathrm{c} \end{gathered}$$

Since speed is less than the speed of light then A will be cause of B.

Part (b) Step : Given information

We have given,

Space-time coordinates of A $(300\mathrm{m},2\mathrm{\mu s})$

Space-time coordinates of C $(2400\mathrm{m},8\mathrm{\mu s})$

we have to find the is A possibly be the cause of C.

SImplify

The distance between the A and C

$$\begin{gathered} ∆\mathrm{x}={\mathrm{x}}_{\mathrm{B}}-{\mathrm{x}}_{\mathrm{A}} \\ ∆\mathrm{x}=2400\mathrm{m}-300\mathrm{m} \\ ∆\mathrm{x}=2100\mathrm{m} \end{gathered}$$

The time taken from A to C

$$\begin{gathered} ∆\mathrm{t}={\mathrm{t}}_{\mathrm{A}}-{\mathrm{t}}_{\mathrm{B}} \\ ∆\mathrm{t}=8\mathrm{\mu s}-2\mathrm{\mu s} \\ ∆\mathrm{t}=6\mathrm{\mu s} \end{gathered}$$

then velocity will be

$$\begin{gathered} \mathrm{V}=\frac{∆\mathrm{x}}{∆\mathrm{t}}=\frac{2100\mathrm{m}}{6\mathrm{\mu s}} \\ \mathrm{V}=350\times {10}^6\mathrm{m}/\mathrm{s} \\ \mathrm{V}>\mathrm{c} \end{gathered}$$

It is not possible.