State the Lorentz transformation for \(x'\) when \(S'\) moves at speed \(v\) along \(+x\) relative to \(S\).

State the Lorentz transformation for \(t'\).

What happens to \(y\) and \(z\) in the standard Lorentz transformation along \(x\)?

In \(S\), an event occurs at \(x=300\,\mathrm m\), \(t=0\). If \(S'\) moves at \(0.600c\), find \(x'\).

In \(S\), an event occurs at \(x=0\), \(t=2.00\,\mu\mathrm s\). If \(S'\) moves at \(0.600c\), find \(t'\).

Write the inverse Lorentz transformations for \(x\) and \(t\).

In \(S\), an event occurs at \(x=900\,\mathrm m\), \(t=4.00\,\mu\mathrm s\). Frame \(S'\) moves at \(0.600c\). Find \(x'\).

For the same event \(x=900\,\mathrm m\), \(t=4.00\,\mu\mathrm s\), and \(v=0.600c\), find \(t'\).

State the relativistic velocity transformation for motion along \(x\).

A particle moves at \(0.900c\) in \(S\). Frame \(S'\) moves at \(0.500c\) in the same direction. Find \(u_x'\).

A spaceship fires a probe forward at \(0.700c\) relative to the ship. The ship moves at \(0.600c\) relative to Earth. Find the probe speed relative to Earth.

Show that the velocity transformation gives \(u_x'=c\) when \(u_x=c\).

Two spacecraft approach each other. In Earth frame, one has velocity \(+0.700c\) and the other \(-0.700c\). What speed does one measure for the other?

An event has \(x'=0\) in a frame moving at \(0.800c\). Use the inverse Lorentz transformation to show its path in \(S\).

Use the Lorentz transformation to find \(x'\) and \(t'\) for \(x=600\,\mathrm m\), \(t=3.00\,\mu\mathrm s\), and \(v=0.800c\).

Derive the time-difference form of the Lorentz transformation from \(t'=\gamma(t-vx/c^2)\).

Show that the Lorentz transformations reduce to Galilean transformations when \(v\ll c\).

Use velocity addition to show that combining two sub-light speeds in the same direction cannot exceed \(c\).

Explain why Lorentz transformations act on events rather than on objects.

Derive the inverse Lorentz transformation by changing the sign of the relative velocity.