In some appropriate units, time $(t)$ and position $(x)$ relation of a moving particle is given by $t=x^2+x$. The acceleration of the particle is

Two cities $X$ and $Y$ are connected by a regular bus service with a bus leaving in either direction every $T$ min. A girl is driving scooty with a speed of $60 \mathrm{~km} / \mathrm{h}$ in the direction $X$ to $Y$ notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction. Choose the correct option for the period $T$ of the bus service and the speed (assumed constant) of the buses.

A particle is moving along $$x$$-axis with its position (x) varying with time $$(t)$$ as $$x=\alpha t^4+\beta t^2+\gamma t+\delta$$. The ratio of its initial velocity to its initial acceleration, respectively, is:

The velocity $$(v)-$$ time $$(t)$$ plot of the motion of a body is shown below:

The acceleration $$(a)-$$ time $$(t)$$ graph that best suits this motion is :

The position of a particle is given by

$$\vec{r}(t)=4 t \hat{i}+2 t^2 \hat{j}+5 \hat{k} $$

where $$\mathrm{t}$$ is in seconds and $$\mathrm{r}$$ in metre. Find the magnitude and direction of velocity $$v(t)$$, at $$t=1 \mathrm{~s}$$, with respect to $$\mathrm{x}$$-axis

A vehicle travels half the distance with speed $$v$$ and the remaining distance with speed $$2 v$$. Its average speed is :

The position-time (x - t) graph for positive acceleration is

The ratio of the distances travelled by a freely falling body in the 1^st, 2^nd, 3^rd and 4^th second

The displacement-time graphs of two moving particles make angles of 30$$^\circ$$ and 45$$^\circ$$ with the x-axis as shown in the figure. The ratio of their respective velocity is