The displacement of a body moving along a straight line is...

The displacement of a body moving along a straight line is given by $x=\mathrm{pt}^{2}+\mathrm{qt}+\mathrm{r}$ where $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are constants and $\mathrm{t}^{\prime}$ represents time. The acceleration of the body will (a) remain a constant (b) decrease (c) increase (d) first increase and then decrease

The displacement of a body moving along a straight line is given by $x=\mathrm{pt}^{2}+\mathrm{qt}+\mathrm{r}$ where $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are constants and $\mathrm{t}^{\prime}$ represents time. The acceleration of the body will
(a) remain a constant
(b) decrease
(c) increase
(d) first increase and then decrease

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Key Concepts

Relationship between Position, Velocity, and Acceleration

In kinematics, the position of a body as a function of time describes its displacement, and its derivatives with respect to time give the velocity and acceleration. Taking the first derivative of the displacement function yields the velocity, and the second derivative yields the acceleration. This hierarchy establishes how changes in displacement over time translate to the body's motion characteristics.

Differentiation in Kinematics

Differentiation is a fundamental concept in calculus used to determine instantaneous rates of change. In kinematics, taking the derivative of a displacement function with respect to time gives the velocity function, and a subsequent derivative gives the acceleration function. This process allows us to analyze how position, velocity, and acceleration are interrelated over time.

Quadratic Displacement and Constant Acceleration

A quadratic displacement function, such as one with a t² term, leads to a first derivative that is a linear function of time (velocity) and a second derivative that is constant (acceleration). This indicates that if the position of a body is described by a quadratic function, its acceleration remains constant, which is a common scenario in uniformly accelerated motion.

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