An astronaut has left the International Space Station to test a new space scooter. Her partner measures the following velocity changes, each taking place in a 10-s interval. What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval? Assume that the positive direction is to the right. (a) At the beginning of the interval, the astronaut is moving toward the right along the x-axis at 15.0 m/s, and at the end of the interval she is moving toward the right at 5.0 m/s. (b) At the beginning she is moving toward the left at 5.0 m/s, and at the end she is moving toward the left at 15.0 m/s. (c) At the beginning she is moving toward the right at 15.0 m/s, and at the end she is moving toward the left at 15.0 m/s.

Verified step by step guidance

1. Identify the initial velocity (v_i) and final velocity (v_f) for each interval, and note the time interval (\(\Delta t\)) which is 10 seconds for all cases.
2. Calculate the change in velocity (\(\Delta v\)) for each interval using the formula \(\Delta v = v_f - v_i\).
3. Determine the average acceleration (a) for each interval using the formula \(a = \frac{\Delta v}{\Delta t}\).
4. Analyze the sign of the average acceleration. If \(\Delta v\) is positive, the acceleration is in the positive direction (to the right). If \(\Delta v\) is negative, the acceleration is in the negative direction (to the left).
5. Determine the magnitude of the average acceleration by taking the absolute value of the calculated acceleration for each interval.

Key Concepts

Acceleration

Acceleration is defined as the rate of change of velocity over time. It can be calculated using the formula a = (v_f - v_i) / Δt, where v_f is the final velocity, v_i is the initial velocity, and Δt is the time interval. Acceleration can be positive or negative, indicating an increase or decrease in speed, respectively. In this context, understanding acceleration is crucial for determining how the astronaut's velocity changes during each interval.

Velocity

Velocity is a vector quantity that describes the speed of an object in a specific direction. It is expressed in units such as meters per second (m/s) and includes both magnitude (how fast) and direction (where to). In the given question, the astronaut's initial and final velocities are essential for calculating the average acceleration. Recognizing the direction of velocity is also important, as it affects the sign of the acceleration.

Direction and Sign Convention

In physics, direction and sign convention are critical for accurately interpreting motion. In this scenario, the positive direction is defined as to the right, which means that velocities and accelerations in that direction are positive, while those to the left are negative. This convention helps in determining the algebraic sign of the average acceleration, which indicates whether the object is speeding up or slowing down in relation to the defined positive direction.