If an object moves with constant acceleration its velocity must

If an object moves with constant acceleration its velocity must

Be constant also

Constant acceleration does not imply that the velocity is constant. In fact, velocity changes continuously when acceleration is constant. Hence, this option is incorrect.

Change by the same amount each second

Constant acceleration means that the change in velocity over each second is constant. Mathematically, this is expressed as

Where:

• $v(t)$ is the velocity at time $t$,
• $v_0$ is the initial velocity,
• $a$ is the constant acceleration.

Thus, the velocity of an object with constant acceleration changes by the same amount each second, making this option correct.

Change by varying amounts depending on its speed

This is incorrect because, under constant acceleration, the change in velocity is not dependent on the object's current speed but rather on the constant rate of acceleration.

Always decrease

Constant acceleration does not necessarily imply a decrease in velocity. Acceleration can be positive or negative:

• Positive acceleration increases the velocity.
• Negative acceleration (also known as deceleration) decreases the velocity.

Therefore, this option is incorrect.

Summary

If an object moves with constant acceleration, its velocity must:

Concept of Constant Acceleration and Linear Velocity Change

Constant acceleration means that an object's acceleration doesn't change with time. This has a direct implication on the velocity of the object.

When acceleration is constant, the velocity of the object changes linearly over time. This relationship can be understood through the following equations derived from the basic principles of kinematics.

For an object with initial velocity $v_0$ and constant acceleration $a$, the velocity at any time $t$ can be expressed as:

This equation shows that velocity $v(t)$ changes linearly with time $t$, because it is a straight-line equation of the form $y = mx + b$, where $m = a$ (the slope, or acceleration) and $b = v_0$ (the initial velocity).

The position $s(t)$ of the object can also be determined from the initial position $s_0$, showing how far the object has moved under constant acceleration. The equation is:

While this equation is quadratic in terms of $t$ for position, the change in velocity remains linear.

To summarize:

• Constant acceleration means that the acceleration $a$ does not change over time.
• This implication is a linear change in velocity, represented by the equation $v(t) = v_0 + at$.

Thus, understanding that constant acceleration results in a linearly changing velocity is crucial in predicting the motion of objects under uniform acceleration.

Understanding the Concept

When an object undergoes constant acceleration $a$, the change in velocity $\Delta v$ depends only on the acceleration and the time duration $t$ for which the acceleration acts. This principle can be mathematically demonstrated using the kinematic equations of motion.

Key Kinematic Equation

The fundamental equation that describes the relationship between initial velocity $v_0$, final velocity $v$, acceleration $a$, and time $t$ under constant acceleration is:

Here,

• $v$ is the final velocity
• $v_0$ is the initial velocity
• $a$ is the constant acceleration
• $t$ is the time duration

Deriving Velocity Change

To find the change in velocity $\Delta v$, we subtract the initial velocity $v_0$ from the final velocity $v$:

Substituting the expression for $v$ from the kinematic equation:

This simplifies to:

Independence from Initial Speed

Notice that the term $v_0$ cancels out in the subtraction. The result $\Delta v = at$ shows that the change in velocity is independent of the initial speed $v_0$. The change in velocity $\Delta v$ depends solely on the constant acceleration $a$ and the time interval $t$.

Practical Implication

This principle is particularly useful in physics and engineering when analyzing motion. For example, whether a car starts from rest or is already moving, if it accelerates at the same constant rate for the same duration, the change in its velocity will be the same, emphasizing the independent nature of velocity change from the initial speed.

Summary: Under constant acceleration, the change in velocity is determined only by the acceleration and the time interval, and it is independent of the initial speed.