Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the acceleration of an object remains constant, then its velocity is constant. b. If the acceleration of an object moving along a line is always 0 then its velocity is constant. c. It is impossible for the instantaneous velocity at all times \(a \leq t \leq b\) to equal the average velocity over the interval \(a \leq t \leq b\) d. A moving object can have negative acceleration and increasing speed.

Short answer

Answer: Yes, a moving object can have negative acceleration and increasing speed if it moves in the negative direction with increasing magnitude. For example, a car going backwards and accelerating in the negative direction.

Step-by-step solution

Check the relationship between acceleration and velocity

Acceleration is defined as the rate of change of velocity. So, if an object has constant acceleration, it means its velocity is changing at a constant rate. In other words, the statement is false.

Provide a counterexample

Think about a car that starts from rest and accelerates uniformly. For example, it could accelerate at 2 m/s². Since it has a constant acceleration, its velocity would increase by 2 m/s every second, so its velocity is not constant. #b. If the acceleration of an object moving along a line is always 0, then its velocity is constant.#

Check the relationship for zero acceleration

If the acceleration of an object is always 0, it means the object's velocity does not change. Since the object's velocity remains the same over time, we can conclude that the statement is true. #c. It is impossible for the instantaneous velocity at all times \(a \leq t \leq b\) to equal the average velocity over the interval \(a \leq t \leq b\)#

Study the relationship between instantaneous and average velocity

The instantaneous velocity is the velocity at a specific point in time, while the average velocity is the total displacement divided by the time interval. It's possible for these two to be equal.

Provide an example

Consider an object moving along a straight line at a constant velocity of 5 m/s for the interval \(0 \leq t \leq 4\). The instantaneous velocity at any point will be 5 m/s, and the average velocity for the interval will be (total displacement) / (time interval) = (5 m/s * 4 s) / 4 s = 5 m/s. So, the statement is false. #d. A moving object can have negative acceleration and increasing speed.#

Understand negative acceleration and increasing speed

Negative acceleration refers to an object slowing down in the positive direction or speeding up in the negative direction. An increasing speed means the magnitude of the object's velocity is growing.

Determine if the statement is true or false

An object can have negative acceleration and increasing speed if it moves in the negative direction with increasing magnitude. For example, a car going backwards and accelerating in the negative direction. This means the statement is true.

Key concepts

Constant Acceleration

Understanding how constant acceleration affects an object's motion is crucial for appreciating Newtonian mechanics. Imagine you're on a roller coaster that starts from the top and accelerates downwards. If this roller coaster maintains constant acceleration, it means that its speed increases by the same amount in each second. Now why isn't the velocity constant? It's because velocity not only considers the speed — how fast an object is moving — but also the direction of that movement.

Let's do a simple activity to grasp the concept better: Drop a ball and consider the force of gravity acting upon it. Gravity ensures the ball's speed increases steadily, causing a constant acceleration which is approximately 9.8 m/s² downwards. This means the ball's velocity is constantly changing because the speed is increasing at a steady rate, even though the acceleration is constant.

Zero Acceleration

Now let's turn our attention to what happens when an object experiences zero acceleration. When acceleration is zero, it means the speed of the object doesn't change — there's no increase or decrease in how fast the object is traveling; however, this doesn't address the direction. Here's a relatable scenario: Picture a car cruising along a highway at a steady 60 miles per hour. The car’s acceleration is zero because its speed is not changing. The velocity in this case is constant, not only because the speed is the same but also because it maintains a straight path, keeping the direction unchanged. This also implies that if the car turned or curved, even while maintaining speed, the velocity would not be constant because velocity accounts for direction.

Instantaneous Velocity

Diving deeper into the nuances of motion, let's explore instantaneous velocity. This term might seem intimidating, but it's actually the speed and direction of an object at a particular instant. To put it simply, if you look at the speedometer of a car at a specific moment, that's the car's instantaneous velocity. Unlike average velocity, which smooths out the entire trip's speed, instantaneous velocity is a snapshot of a single moment in time.

For instance, while you travel from home to school, you may stop at red lights, accelerate, or decelerate. Regardless of these changes, at any given point, if you were to check your speedometer, you'd see your instantaneous velocity. Students can sometimes get confused here — remember that if an object moves at a consistent speed in a straight direction for any duration, its instantaneous velocity doesn't vary; it remains constant.

Average Velocity

Lastly, when looking at an object's journey as a whole, we can calculate its average velocity. This value doesn't consider the various changes in speed or pauses — it just tells us the overall change in position per unit of time.

To imagine this, think about running around a track. You complete one lap at varying speeds, sometimes sprinting, sometimes jogging. The average velocity is found by dividing the total distance traveled by the time it takes to complete the lap. If you were to jog at a steady pace without changing speed or stopping once, your instantaneous velocity at every point around the track would align perfectly with your average velocity. But in most real-world scenarios, because of varying speeds and possible changes in direction, instantaneous and average velocity will differ.