An object's velocity is \(\vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath},\) where \(t\) is time and \(c\) and \(d\) are positive constants with appropriate units. What's the direction of the object's acceleration?

When we discuss the motion of an object, its velocity vector plays a critical role in understanding how it is moving. The velocity vector, denoted as \( \textbf{v} \) or \( \vec{v} \), is a mathematical representation that combines both the speed at which an object is moving and the direction in which it is moving.

In the given exercise, the velocity vector is expressed as \( \vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath} \). This tells us that the object's velocity depends on time, with \( t^{3} \) influencing the motion in the 'x' direction (represented by \( \hat{\imath} \) unit vector) and a constant velocity component \( d \) in the 'y' direction (represented by \( \hat{\jmath} \) unit vector).

It's important to distinguish that this velocity is not constant. The object's motion in the 'x' direction changes over time due to the \( t^{3} \) factor, directly impacting the object's acceleration.

Acceleration can be comprehensively understood through its relationship with velocity by using calculus. Specifically, acceleration is defined as the derivative of velocity with respect to time. Mathematically, this is shown as \( \vec{a} = \frac{\mathrm{d}\vec{v}}{\mathrm{d}t} \).

Derivation involves finding how a function changes at any given moment, which, for motion, translates to how quickly velocity changes over time. If an object's velocity is changing, it's accelerating, and this rate of change is captured by the acceleration vector.

In our example, taking the time derivative of the velocity vector \( \vec{v}=c t^{3} \hat{\imath}+d \hat{\jmath} \) results in \( \vec{a}= 3ct^{2}\hat{\imath}+0\hat{\jmath} \). The derivative of the constant term \( d \hat{\jmath} \) is zero because a constant's rate of change over time is always zero.

The acceleration vector, similar to the velocity vector, is a vector that describes the rate at which an object's velocity changes over time. For an object moving in two dimensions, its acceleration vector has two components, corresponding to the 'x' and 'y' directions.

The acceleration vector derived in the previous section, \( \vec{a}= 3ct^{2}\hat{\imath}+0\hat{\jmath} \), has a non-zero component only in the \( \hat{\imath} \) direction, while the \( \hat{\jmath} \) component is zero. This indicates that the object is accelerating only in the 'x' direction, and its 'y' component of velocity remains constant (as no acceleration implies no change in velocity).

Knowing the acceleration vector is crucial for predicting future positions and velocities of moving objects, especially in physics problems where kinematics is involved.

Vectors are mathematical entities that have both magnitude and direction. These vectors can be split into their respective vector components, which represent the influences in perpendicular directions. For a 2-dimensional vector, these components are usually along the 'x' axis (\( \hat{\imath} \) direction) and the 'y' axis (\( \hat{\jmath} \) direction).

Components are used because they simplify the analysis of motion by allowing us to look at each dimension independently. For instance, in our problem, the velocity vector's components are \( c t^{3} \) in the 'x' direction and \( d \) in the 'y' direction. When we calculated acceleration, we treated each component separately, leading to an acceleration that only affects the 'x' component due to the nature of the velocity vector's time dependency (\( t^{3} \)).

Using vector components is foundational in physics and engineering, as it simplifies complex problems into more manageable parts.