Question

If the path of a moving point is the curve $\mathrm{x}=\mathrm{at}$, $\mathrm{y}=\mathrm{bsinat}$, then its acceleration at any instant

• A. Is constant
• B. Varies as the distance from the axis of $\mathrm{x}$
• C. Varies as the distance from the axis of $\mathrm{y}$
• D. Varies as the distance of the point from the origin

Answer 1

The correct option is C

Varies as the distance from the axis of $\mathrm{y}$

Step 1: Given data

The path of a moving point is the curve $\mathrm{x}=\mathrm{at}$ and $\mathrm{y}=\mathrm{bsinat}$

Step 2: Formula used

Velocity is the first derivative of the displacement,

That is, $\mathrm{v}=\frac{\mathrm{ds}}{\mathrm{dt}}$

Acceleration is the second derivative of the displacement, in other words, acceleration is the derivative of velocity

That is, $\mathrm{a}=\frac{{\mathrm{d}}^2\mathrm{s}}{{\mathrm{dt}}^2}$ or $\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}$

Step 3 Velocity and acceleration from the axis of $\mathrm{x}$

The path of a moving point is the curve $\mathrm{x}=\mathrm{at}$ and $\mathrm{y}=\mathrm{bsinat}$

For $\mathrm{x}=\mathrm{at}$,

The velocity component along x-axis is given by,

$$\begin{gathered} {\mathrm{v}}_{\mathrm{x}}=\frac{\mathrm{dx}}{\mathrm{dt}} \\ {\mathrm{v}}_{\mathrm{x}}=\frac{\mathrm{d}\left(\mathrm{at}\right)}{\mathrm{at}} \\ {\mathrm{v}}_{\mathrm{x}}=\mathrm{a} \end{gathered}$$

Accelaration is given by,

$$\begin{gathered} {\mathrm{a}}_{\mathrm{x}}=\frac{d{\mathrm{v}}_{\mathrm{x}}}{\mathrm{dt}} \\ {\mathrm{a}}_{\mathrm{x}}=\frac{\mathrm{d}\left(\mathrm{a}\right)}{\mathrm{dt}} \\ {\mathrm{a}}_{\mathrm{x}}=0 \end{gathered}$$

Thus, the acceleration does not vary with the change in the distance along the x-axis.

Step 4: Velocity and acceleration from the axis of $\mathrm{y}$

For $\mathrm{y}=\mathrm{bsinat}$,

Velocity component along the y-axis is given by,

$$\begin{gathered} {\mathrm{v}}_{\mathrm{y}}=\frac{\mathrm{dy}}{\mathrm{dt}} \\ {\mathrm{v}}_{\mathrm{y}}=\frac{\mathrm{d}\left(\mathrm{bsinat}\right)}{\mathrm{dt}} \\ {\mathrm{v}}_{\mathrm{y}}=\mathrm{bacosat} \end{gathered}$$

Accelaration is given by,

$$\begin{gathered} {\mathrm{a}}_{\mathrm{y}}=\frac{{\mathrm{dv}}_{\mathrm{y}}}{\mathrm{dt}} \\ {\mathrm{a}}_{\mathrm{y}}=\frac{\mathrm{d}\left(\mathrm{bacosat}\right)}{\mathrm{dt}} \\ {\mathrm{a}}_{\mathrm{y}}=-{\mathrm{ba}}^2\sin \mathrm{at} \\ {\mathrm{a}}_{\mathrm{y}}=-a^{2}y \end{gathered}$$

As the value of acceleration along the y-axis depends upon the distance along y-axis, therefore, ${\mathrm{a}}_{\mathrm{y}}$ changes as $\mathrm{y}$ changes.

Hence, option C is the correct answer.