The position of a particle is given by; \(\vec{r}=(3.0t\hat{i}-2.0t^{2}\hat{j}+4.0\hat{k})\) m $\stackrel{}{\mathrm{}}$
 where \(t\) is in seconds and the coefficients have the proper units for \(r\) to be in meters. The magnitude and direction of \(\vec{v}(t)\) at \(t=1.0\) s are:

1.	\(4\) m/s \(53^\circ\) with x-axis
2.	\(4\)  m/s \(37^\circ\) with x-axis
3.	\(5\) m/s \(53^\circ\) with y-axis
4.	\(5\)  m/s \(53^\circ\) with x-axis

A particle starts from the origin at t = 0 with a velocity of 5.0î m/s and moves in the x-y plane under the action of a force that produces a constant acceleration of (3.0î +2.0 ĵ ) $\mathrm{m}/{\mathrm{s}}^2$. What is the speed of the particle at the instant its x-coordinate is 84 m?

Two particles are separated by a horizontal distance x as shown in the figure. They are projected at the same time as shown in the figure with different initial speeds. The time after which the horizontal distance between them becomes zero will be:
                     
1.  $\frac{x}{u}$

2.  $\frac{u}{2x}$

3.  $\frac{2u}{x}$

4.  None of the above

Two boys are standing at the ends \(A\) and \(B\) of the ground where \(AB =a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy at \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:

A stone tied to the end of a 1 m long string is whirled in a horizontal circle at a constant speed. If the stone makes 22 revolutions in 44 seconds, what is the magnitude and direction of acceleration of the stone?

A car is moving at a speed of 40 m/s on a circular track of radius 400 m. This speed is increasing at the rate of 3 m/s². The acceleration of the car is:

1. 4 m/s²

2. 7 m/s²

3. 5 m/s²

4. 3 m/s²

When a particle is projected at some angle to the horizontal, it has a range R and time of flight t₁. If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is t₂, then:

1. ${\mathrm{t}}_1+{\mathrm{t}}_2=\frac{2\mathrm{R}}{\mathrm{g}}$

2. ${\mathrm{t}}_1-{\mathrm{t}}_2=\frac{\mathrm{R}}{\mathrm{g}}$

3. ${\mathrm{t}}_1{\mathrm{t}}_2=\frac{2\mathrm{R}}{\mathrm{g}}$

4. ${\mathrm{t}}_1{\mathrm{t}}_2=\frac{\mathrm{R}}{\mathrm{g}}$

A person, reaches a point directly opposite on the other bank of a flowing river, while swimming at a speed of 5 m/s at an angle of \(120^\circ\) with the flow. The speed of the flow must be:
1. \(2.5\) m/s
2. \(3\) m/s
3. \(4\) m/s
4. \(1.5\) m/s

Consider the motion of the tip of the second hand of a clock. In one minute (assuming \(R\) to be the length of the second hand), its:

A particle starts from the origin at t = 0 sec with a velocity of $10$ $\hat{j}$ $m/s$ and moves in the x-y plane with a constant acceleration of \((8.0\hat i +2.0 \hat j)\) \(\text{ms}^{-2}\). At what time is the x- coordinate of the particle 16 m?