The problem you've uploaded asks:

"A rocket blasts off from rest and attains a speed of 56.2 m/s in 13.4 s. An astronaut has a mass of 65.3 kg. What is the astronaut's apparent weight during takeoff?"

Let's break this down:

1. The initial speed of the rocket is $v_0 = 0 \, \text{m/s}$ (since it blasts off from rest).
2. The final speed of the rocket after 13.4 seconds is $v_f = 56.2 \, \text{m/s}$.
3. Time $t = 13.4 \, \text{s}$.
4. The astronaut's mass $m = 65.3 \, \text{kg}$.
5. We need to find the astronaut's apparent weight during takeoff, which depends on both the gravitational force and the force due to acceleration.

Step 1: Calculate the acceleration of the rocket

Using the formula for acceleration:

$a = \frac{v_f - v_0}{t}$

Substitute the known values:

$a = \frac{56.2 \, \text{m/s} - 0 \, \text{m/s}}{13.4 \, \text{s}} = \frac{56.2}{13.4}$

Let me calculate the exact value.The acceleration of the rocket is approximately $4.19 \, \text{m/s}^2$.

Step 2: Calculate the astronaut's apparent weight

The apparent weight is the sum of the normal gravitational weight and the additional force from the rocket's acceleration.

1. The gravitational force is $F_{\text{gravity}} = m \cdot g$, where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.
2. The additional force due to the rocket's acceleration is $F_{\text{rocket}} = m \cdot a$.

Thus, the apparent weight $F_{\text{apparent}}$ is:

$F_{\text{apparent}} = m \cdot (g + a)$

Substitute the values:

$F_{\text{apparent}} = 65.3 \cdot (9.8 + 4.19)$

Let's calculate this next.The astronaut's apparent weight during takeoff is approximately $913.8 \, \text{N}$ (Newtons).

Would you like more details on this, or do you have any further questions?

Here are 5 related questions to deepen your understanding:

1. How does the apparent weight change if the rocket's acceleration increases?
2. What would be the astronaut's apparent weight if the rocket's acceleration was zero?
3. How does the apparent weight during takeoff compare to the astronaut's true weight on Earth?
4. What happens to the astronaut's apparent weight if the rocket accelerates downward?
5. How would this calculation differ in a location with a different gravitational constant, such as the Moon?

Tip: Apparent weight increases when an object is accelerating upwards because of the added force due to the acceleration.