A particle moving in the xy plane experiences | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given force $\overrightarrow{F} = m\overrightarrow{a} = k(v_y \hat{i} + v_x \hat{j})$.Thus,$a_x = \frac{dv_x}{dt} = \frac{k}{m} v_y$ and $a_y = \f

Vedclass · Accepted Answer

The quantity $\overrightarrow{v} 	imes \overrightarrow{a}$ is constant in time.

Vedclass · Answer

(C) Given force $\overrightarrow{F} = m\overrightarrow{a} = k(v_y \hat{i} + v_x \hat{j})$.Thus,$a_x = \frac{dv_x}{dt} = \frac{k}{m} v_y$ and $a_y = \frac{dv_y}{dt} = \frac{k}{m} v_x$.Dividing the two equations: $\frac{dv_y}{dv_x} = \frac{v_x}{v_y} \implies v_y dv_y = v_x dv_x$.Integrating both sides: $\int v_y dv_y = \int v_x dv_x \implies v_y^2 = v_x^2 + C \implies v_y^2 - v_x^2 = 	ext{constant}$.Now,calculate $\overrightarrow{v} 	imes \overrightarrow{a}$:$\overrightarrow{v} 	imes \overrightarrow{a} = (v_x \hat{i} + v_y \hat{j}) 	imes \frac{k}{m}(v_y \hat{i} + v_x \hat{j})$$= \frac{k}{m} [v_x^2 (\hat{i} 	imes \hat{j}) + v_y^2 (\hat{j} 	imes \hat{i})]$$= \frac{k}{m} [v_x^2 \hat{k} - v_y^2 \hat{k}] = \frac{k}{m} (v_x^2 - v_y^2) \hat{k}$.Since $v_y^2 - v_x^2$ is constant,$\overrightarrow{v} 	imes \overrightarrow{a}$ is constant in time.

Similar Questions

Two particles are projected from the same point with the same speed at different angles $\theta _1$ and $\theta _2$ to the horizontal. They have the same range. Their times of flight are $t_1$ and $t_2$ respectively.

$A$ particle at a distance of $1 \ m$ from the origin starts moving,such that $dr/d\theta = r$,where $r$ and $\theta$ are polar coordinates. Then,the angle between the resultant velocity and the tangential velocity is

Starting from the origin at time $t=0,$ with initial velocity $5 \hat{j} \, m/s,$ a particle moves in the $x-y$ plane with a constant acceleration of $(10 \hat{i} + 4 \hat{j}) \, m/s^2$. At time $t$,its coordinates are $(20 \, m, y_0 \, m)$. The values of $t$ and $y_0$ are respectively:

Vedclass Test Series

Exam Paper Generator

Online Exam Module