Question

directions: answer the following questions using complete sentences.

1. evaluate “roller coasters scare me, but only when they are accelerating,” a friend says. “once they start slowing down, i’m fine.” how would you respond to this statement?

1. analyze imagine a basketball bouncing. as it is moving up, does it have a positive or a negative acceleration? why?

1. analyze now imagine the same bouncing basketball as it is moving down. does it have a positive or a negative acceleration? why?

Explanation:

Question 1

To respond, we need to correct the friend's misunderstanding about acceleration. Acceleration is the rate of change of velocity, which includes changes in speed and direction. Roller coasters have acceleration during slowing down (deceleration, a type of acceleration) and when changing direction (like in loops or curves). So the friend is incorrect because slowing down is also acceleration (negative acceleration if we take the direction of motion as positive), and direction changes also involve acceleration. A proper response would explain that acceleration isn't just speeding up; slowing down (deceleration) and direction changes are also acceleration. For example: "Your statement has a misunderstanding. Acceleration is any change in velocity, including slowing down (deceleration) and changing direction. Roller coasters often accelerate when slowing down (negative acceleration relative to the direction of motion) and also when changing direction (like in loops), so you still experience acceleration when they slow down or change direction."

To determine the acceleration of the basketball moving up, we consider the definition of acceleration (change in velocity over time) and the force acting on it (gravity). When the basketball moves up, its velocity is in the upward direction, but gravity acts downward, causing the velocity to decrease (slow down). If we take the upward direction as positive, the change in velocity (\(\Delta v\)) is negative (since final velocity is less than initial velocity). Acceleration \(a=\frac{\Delta v}{\Delta t}\), so with \(\Delta v\) negative and \(\Delta t\) positive, acceleration is negative.

For the basketball moving down, we again use the definition of acceleration and the force of gravity. When the basketball moves down, its velocity is in the downward direction. Gravity acts downward, causing the velocity to increase (the ball speeds up as it falls down). If we take the downward direction as positive (or even if we take upward as positive, the change in velocity will be positive relative to the direction of motion here), the change in velocity (\(\Delta v\)) is positive (final velocity > initial velocity, as it speeds up while moving down). Acceleration \(a=\frac{\Delta v}{\Delta t}\), so with \(\Delta v\) positive and \(\Delta t\) positive, acceleration is positive (if we take downward as positive) or, if we take upward as positive, the velocity is negative and becoming more negative (increasing in magnitude downward), so \(\Delta v\) is negative - wait, no, let's clarify. Let's define upward as positive. Initial velocity when moving down is negative (downward), and as it falls, it speeds up, so final velocity is more negative (e.g., from -2 m/s to -5 m/s). So \(\Delta v = -5 - (-2) = -3\) m/s? Wait, no, that's a mistake. Wait, when moving down, the velocity is in the negative direction (if upward is positive), and it's increasing in magnitude (speeding up). So the change in velocity: if initial velocity \(v_i=-2\) m/s (downward), final velocity \(v_f = -5\) m/s (more downward), then \(\Delta v = v_f - v_i = -5 - (-2) = -3\) m/s. But acceleration due to gravity is downward, so if upward is positive, acceleration is negative? Wait, no, I messed up. Let's use the correct approach: acceleration is the rate of change of velocity. When the ball is moving down, the force of gravity is in the same direction as the motion (downward), so the velocity is increasing in the downward direction. If we take downward as the positive direction, then the velocity is positive and increasing, so \(\Delta v\) is positive, and acceleration \(a=\frac{\Delta v}{\Delta t}\) is positive. If we take upward as positive, velocity is negative and becoming more negative (e.g., from -1 m/s to -3 m/s), so \(\Delta v = -3 - (-1) = -2\) m/s, and \(a=\frac{-2}{\Delta t}\), which is negative? Wait, no, that's conflicting with gravity. Wait, no, the acceleration due to gravity is always downward, so regardless of the direction of motion, the acceleration is downward. So when moving up, acceleration is downward (opposite to motion, negative if upward is positive). When moving down, acceleration is downward (same as motion, so if downward is positive, acceleration is positive; if upward is positive, acceleration is negative? Wait, no, let's use the standard: acceleration is positive if it's in the direction of the chosen positive axis. Let's choose upward as positive. The acceleration due to gravity is \(g = -9.8\) m/s² (downward). When the ball is moving down, its velocity is negative (downward), and it's speeding up, so the velocity is becoming more negative (e.g., from -2 m/s to -5 m/s in 1 second). So \(\Delta v = -5 - (-2) = -3\) m/s, so \(a=\frac{\Delta v}{\Delta t}=\frac{-3}{1}=-3\) m/s²? But that's the same as gravity. Wait, no, the key is: when moving down, the acceleration is in the same direction as the motion (downward), so if we take downward as positive, acceleration is positive. Let's redefine positive as downward. Then velocity is positive, and as it falls, velocity increases (e.g., from 2 m/s to 5 m/s), so \(\Delta v = 3\) m/s, \(a=\frac{3}{\Delta t}\) is positive. So the correct way is: when the basketball moves down, the acceleration is…

Answer:

Your statement has a misunderstanding. Acceleration is any change in velocity, including slowing down (deceleration) and changing direction. Roller coasters often accelerate when slowing down (negative acceleration relative to the direction of motion) and also when changing direction (like in loops), so you still experience acceleration when they slow down or change direction.

Question 7