A rocket blasts off from rest and attains a speed of 53.8 m/s in 17.4 s. An astronaut has a mass of 67.2 kg. What is the astronaut's apparent weight during takeoff?
To calculate the astronaut's apparent weight during takeoff, we need to use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
In this case, the force we're interested in is the apparent weight of the astronaut during takeoff. We can calculate this by subtracting the force due to gravity from the total force acting on the astronaut.
The force due to gravity can be calculated using the formula F = mg, where m is the mass of the astronaut and g is the acceleration due to gravity (9.8 m/s^2).
First, let's calculate the total force acting on the astronaut during takeoff.
Using the formula v = u + at, where v is the final velocity (53.8 m/s), u is the initial velocity (0 m/s), a is the acceleration, and t is the time (17.4 s), we can rearrange the formula to solve for acceleration.
a = (v - u) / t
a = (53.8 m/s - 0 m/s) / 17.4 s
a = 3.087 m/s^2
Now, let's calculate the total force acting on the astronaut using Newton's second law of motion.
F = ma
F = 67.2 kg * 3.087 m/s^2
F ≈ 206.99 N
Next, let's calculate the force due to gravity.
F_gravity = mg
F_gravity = 67.2 kg * 9.8 m/s^2
F_gravity ≈ 658.56 N
Finally, to find the apparent weight of the astronaut during takeoff, we subtract the force due to gravity from the total force acting on the astronaut.
Apparent weight = Total Force - Force due to gravity
Apparent weight ≈ 206.99 N - 658.56 N
Apparent weight ≈ -451.57 N
The negative sign indicates that the astronaut's apparent weight during takeoff is actually less than his or her actual weight due to the upward acceleration of the rocket.