An astronaut (mass of 100 kg, with equipment) is headed back to her space station at a speed of 0.750 m/s but at the wrong angle. To correct her direction, she fires rockets from her backpack at right angles to her motion for a brief time. These directional rockets exert a constant force of 100.0 N for only 0.200 s. [Neglect the small loss of mass due to burning fuel and assume the impulse is at right angles to her initial momentum.] (a) What is the magnitude of the impulse delivered to the astronaut? (b) What is her new direction (relative to the initial direction)? (c) What is her new speed?

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To solve this problem, we can use the principle of conservation of momentum. The impulse delivered by the rockets will change the momentum of the astronaut, enabling her to change direction. Let's calculate the answers step-by-step:

(a) What is the magnitude of the impulse delivered to the astronaut?

Impulse is defined as the product of force and time:

Impulse = force × time

Given:
Force = 100.0 N
Time = 0.200 s

Impulse = 100.0 N × 0.200 s
Impulse = 20.0 N s

Therefore, the magnitude of the impulse delivered to the astronaut is 20.0 N s.

(b) What is her new direction (relative to the initial direction)?

Since the impulse is delivered at right angles to the astronaut's initial momentum, the new direction will be perpendicular to her initial direction. In other words, her new direction will be 90° to the left or right of her initial direction. The problem statement doesn't provide enough information to determine the specific left or right direction.

(c) What is her new speed?

To find the change in speed of the astronaut, we need to use the principle of conservation of momentum. The initial momentum of the astronaut will be equal to the final momentum since no external forces are acting on the system (neglecting air resistance and small mass loss).

Initial momentum = Final momentum

Since the impulse is delivered perpendicular to the initial momentum, there will be no change in the magnitude of momentum. Therefore, the new speed of the astronaut will be the same as her initial speed.

Hence, her new speed will remain at 0.750 m/s.

To answer these questions, we'll first need to understand the concept of impulse and its relation to the change in momentum. Impulse is defined as the product of force and the time interval over which it acts, and it is equal to the change in momentum of an object. Mathematically, impulse (J) can be calculated as J = F * Δt, where F is the force applied to the object and Δt is the time interval.

Let's now proceed to answer each part of the question:

(a) What is the magnitude of the impulse delivered to the astronaut?

In this case, the astronaut fires rockets from her backpack, exerting a constant force of 100.0 N for a time duration of 0.200 s. Therefore, the magnitude of the impulse can be calculated using the equation J = F * Δt. Plugging in the given values, we get:

J = (100.0 N) * (0.200 s)
J = 20.0 N⋅s

Therefore, the magnitude of the impulse delivered to the astronaut is 20.0 N⋅s.

(b) What is her new direction (relative to the initial direction)?

Since the impulse is at right angles to the astronaut's initial momentum, it causes a change in the direction of the momentum vector. The new direction will be perpendicular (at right angles) to the initial direction.

(c) What is her new speed?

The question provides the initial speed of the astronaut, which is 0.750 m/s. Since the impulse only causes a change in direction and not magnitude (speed), the new speed of the astronaut will be the same as the initial speed. Therefore, her new speed remains at 0.750 m/s.