• University of Toronto

• PHYA 21

PHYA21 - Physics II (Physical Sciences)

Practical Session #11

Week #12: Wednesday, Apr.01 - Friday, Apr.03

Reading Material: Chapter 37 (37.1 - 37.6)

1. Sound waves travel at a speed of 340 m/s relative to air.

You pursue a wave of sound at a speed that is 99% of the speed of sound relative to air.

(a) Find the speed of the sound wave relative to you.

(b) What is the difference between the speed of the sound wave relative to you and a speed of exactly 1%

of the speed of sound relative to air? That is, how different is the result derived using the relativistic

addition of velocities, from the result found using the Galilean addition of velocities?

2. Consider frame S′ moving along the positive x direction with speed v relative to S. We know how to transform between the x and y coordinates on S, to the x′ and y′ coordinates on S′ (see Eq. 37-21 and Eq. 37-22).

Similarly, we know how to transform between the horizontal component of the speed of an object on

S given by ux, and the horizontal component of the speed of an object on S′ given by u′ x (see Eq. 37-29).

(a) Derive the respective Lorentz transformations for t and t′ presented in Eq. 37-21 and Eq. 37-22.

Hint: The procedure was outlined in Lecture 22 starting around timestamp 15:51.

(b) Show that uy, the vertical component of the speed of an object relative to S, transforms to u′ y relative

to S′ according to:

u′

y =

u

y

γv (1 - uxv/c2),

where ux is the horizontal component of the velocity of the object relative to S, and γv is the relativistic factor between the two frames S and S′.

3. A spaceship travels at 0.8c relative to Earth, and is able to fire projectiles at 0.6c relative to the spaceship.

(a) If the spaceship moves towards Earth and fires the projectile directly towards Earth, how fast do

observers on Earth see the projectile approaching?

(b) If the spaceship moves away from Earth and fires the projectile directly towards Earth, how fast is the

projectile moving relative to Earth? Is it approaching Earth or moving away from Earth?

(c) The spaceship is passing by the Earth at a distance of a few light-minutes. The spaceship fires the

projectile towards Earth and perpendicular to its current direction of travel. What is the speed of the

projectile relative to Earth? What is the direction of the projectile as seen by Earth?

Hint: You might want review the result from Activity 2(b)