Which of these statements about acceleration are true

Analysis of Statements about Acceleration

Let's evaluate the given statements to determine their validity.

Statement (a): A car must always have an acceleration in the same direction as its velocity

This statement is incorrect. A car can have an acceleration in a direction opposite to its velocity, which typically leads to a decrease in its speed (deceleration). For example, when a car is moving forward but the driver applies the brakes, the acceleration is directed opposite to the direction of the car's velocity.

Statement (b): It's possible for a slowing car to have a positive acceleration

This statement is true depending on the chosen frame of reference. In physics, acceleration is a vector quantity that depends on the coordinate system. If we consider a positive acceleration to be in the opposite direction of the velocity (assuming we are moving in the negative direction), the car will slow down.

In this case, the car has a negative velocity (moving backward) but a positive acceleration, causing it to slow down in the backward direction.

Statement (c): An object with constant nonzero acceleration can never stop and remain at rest

This statement is true. If an object has a constant nonzero acceleration, this implies that it is continually changing its velocity. Therefore, it cannot stop and remain at rest because remaining at rest would mean having zero velocity and zero acceleration.

Important note: The constant acceleration ensures that the object transitions through various velocity states without pausing.

Given these evaluations, we conclude that statement (b) and statement (c) are true, while statement (a) is false.

Understanding the Velocity and Acceleration Relationship

The relationship between velocity and acceleration is fundamental in physics, particularly in the study of motion.

Definitions

1. Velocity ($\vec{v}$) is the rate at which an object changes its position. It is a vector quantity, meaning it has both magnitude and direction.
2. Acceleration ($\vec{a}$) is the rate at which an object changes its velocity. Like velocity, it is also a vector quantity.

Mathematical Relationship

The relationship between velocity and acceleration is described by the following differential equation:

This means that acceleration is the time derivative of velocity.

Integral Form

Conversely, if you know the acceleration, you can find the velocity by integrating with respect to time:

Here, $\vec{v}_0$ is the initial velocity at time $t_0$.

Constant Acceleration

In cases of constant acceleration, the relationship simplifies to:

Where:

• $\vec{v}$ is the final velocity,
• $\vec{v}_0$ is the initial velocity,
• $\vec{a}$ is the constant acceleration, and
• $t$ is the time elapsed.

Key Points

• Acceleration indicates how quickly the velocity of an object is changing.
• A positive acceleration means an increase in velocity, while a negative acceleration (often called deceleration) means a decrease in velocity.
• The magnitude of velocity is known as speed, while acceleration can change the speed and/or direction of motion.

Understanding this relationship is crucial for solving problems in classical mechanics, such as projectile motion, free fall, and circular motion.

Explanation

Deceleration refers to the reduction in speed or velocity of an object. It is essentially negative acceleration. In physics, acceleration ($a$) is defined as the rate of change of velocity ($v$) with respect to time ($t$):

When an object slows down, its acceleration is negative, and we call this deceleration. For example, if a car is slowing down from a speed of 60 km/h to 10 km/h, it is experiencing deceleration.

Frame of reference

A frame of reference is a coordinate system or a set of axes within which to measure the position, orientation, and other properties of objects in it. The choice of frame of reference affects how we describe the motion of objects. It can be inertial (non-accelerating) or non-inertial (accelerating).

Example

Consider a car decelerating in two different frames of reference:

1. Inertial Frame of Reference: Suppose an observer is standing on the sidewalk as the car decelerates. The car’s velocity and the observer’s reference frame are not accelerating, so the observer directly measures the car's deceleration.

2. Non-Inertial Frame of Reference: If we observe the car from another vehicle that is also decelerating, the relative deceleration will be different. Let the deceleration of the second vehicle be $a_{\text{second}}$. The observed deceleration of the car from this frame can be given by:

Key Point: The perceived deceleration can vary depending on the observer’s frame of reference.

Conclusion

Understanding deceleration and frame of reference is crucial in physics as it allows us to describe motion accurately from different perspectives. When objects decelerate, their rate of slowing down can look different depending on whether you are observing from a stationary point or from an accelerating/decelerating frame.