(a) Determine the relativistic momentum of the object. (b) Determine the total relativistic energy, in joules, of the object according to a stationary observer.

Steps to Solve

1. Calculate the Lorentz Factor ($\gamma$)

The Lorentz factor is given by the formula:

$$ \gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}} $$

Here, $v = 0.81c$. Substituting this in:

$$ \gamma = \frac{1}{\sqrt{1 - (0.81)^2}} = \frac{1}{\sqrt{1 - 0.6561}} = \frac{1}{\sqrt{0.3439}} \approx 1.708 $$

2. Calculate the Relativistic Momentum ($p$)

The formula for relativistic momentum is:

$$ p = \gamma m v $$

Substituting the known values ($m = 1.2 , \text{kg}$, $v = 0.81c$, and $\gamma \approx 1.708$):

$$ p = (1.708)(1.2)(0.81c) \approx 1.688 , kg \cdot m/s \enspace \text{(after substituting $c \approx 3 \times 10^8 , m/s$)} $$

Calculating $p$:

$$ p \approx (1.708)(1.2)(0.81) \times 3 \times 10^8 \approx 41.4 \times 10^7 , kg \cdot m/s \approx 4.14 \times 10^8 , kg \cdot m/s $$

3. Calculate Total Relativistic Energy ($E$)

The total relativistic energy can be calculated using the formula:

$$ E = \gamma mc^2 $$

Substituting in the values:

$$ E = (1.708)(1.2)(c^2) = (1.708)(1.2)( (3 \times 10^8)^2 ) $$

Calculating further:

$$ E = (1.708)(1.2)(9 \times 10^{16}) \approx 2.54 \times 10^{17} , J $$