A rocket blasts off from rest and attains a speed of 42.2 m/s in 10.9 s. An astronaut has a mass of 63.8 kg. What is the astronaut's apparent weight during takeoff?

To determine the astronaut's apparent weight during takeoff, we need to consider the forces acting on the astronaut.

1. First, calculate the astronaut's weight:
The weight of an object is given by the formula: weight = mass x gravitational acceleration.

Given:
Mass of the astronaut (m) = 63.8 kg.

Gravitational acceleration (g) can be taken as 9.8 m/s^2.

Weight = 63.8 kg * 9.8 m/s^2 = 625.24 N

Therefore, the astronaut's weight is 625.24 Newtons.

2. Next, consider the forces acting during takeoff:
During takeoff, the rocket exerts an upward force on the astronaut. This upward force is equal to the weight of the astronaut plus any additional force required to accelerate the astronaut.

Using Newton's second law of motion, F = m * a, where F is the total force acting on the astronaut, m is the mass of the astronaut, and a is the acceleration.

In this case, the total force F is the sum of the weight and the additional force provided by the rocket.

3. Calculate the additional force provided by the rocket:
We can use the formula: force = mass x acceleration.

The mass of the rocket is not provided, but we can use the relationship between force, mass, and acceleration to find the additional force acting on the astronaut.

Acceleration = (Final speed - Initial speed) / Time

Given:
Initial speed (u) = 0 (since the rocket starts from rest)
Final speed (v) = 42.2 m/s
Time (t) = 10.9 s

Acceleration = (42.2 m/s - 0 m/s) / 10.9 s = 3.87 m/s^2 (approx.)

Now, we can calculate the additional force:

Force = mass of the astronaut x acceleration of the astronaut
= 63.8 kg x 3.87 m/s^2
= 245.706 N (approx.)

Therefore, the additional force provided by the rocket is approximately 245.706 Newtons.

4. Calculate the apparent weight of the astronaut:
The apparent weight of the astronaut is the total force acting on the astronaut during takeoff.

Apparent weight = Weight + Additional force
= 625.24 N + 245.706 N
= 870.946 N (approx.)

Therefore, the astronaut's apparent weight during takeoff is approximately 870.946 Newtons.