A particle starts from the origin at t = 0 s with the velocity of $$10 \; \hat{j} \; m \; s^{-1}$$ and moves in the x –y plane with a constant acceleration of $$\left ( 8.0 \; \hat{i} + 2.0 \; \hat{j}\right ) \; m \; s^{-2}$$

(a) At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?

(b) What is the speed of the particle at the time?

Asked by Abhisek | 1 year ago |  80

##### Solution :-

(a) Velocity of the particle =$$10 \; \hat{j} \; m \; s^{-1}$$

Acceleration of the particle = $$\left (8.0 \; \hat{i} + 2.0 \; \hat{j}\right) \; m \; s^{-2}$$

But, $$\vec{a} = \dfrac{\vec{dv}}{dt} = 8.0 \; \hat{i} + 2.0 \; \hat{j}\vec{dv} = \left (8.0 \; \hat{i} + 2.0 \; \hat{j} \right )\ ;dt$$

Integrating both the sides:

$$\vec{v}\left ( t \right ) = 8.0 t \; \hat{i} + 2.0 t \; \hat{j} + \vec{u}$$

Where,$$\vec{u}$$ = velocity vector of the particle at t = 0

$$\vec{v}$$= velocity vector of the particle at time t

But, $$\vec{v} = \dfrac{\vec{dr}}{dt}[/latex]\vec{dr} = \vec{v}\; dt$$

=$$\left (8.0 t \; \hat{i} + 2.0 t \; \hat{j} + \vec{u} \right ) \; dt$$

Integrating the equations with the conditions:

At t = 0; r = 0 and at t = t; r = r.

$$\vec{r} = \vec{u}t + \frac{1}{2}8.0t^{2} \; \hat{i} + \frac{1}{2} \times 2.0t^{2} \; \hat{j}\vec{r}$$

$$= \vec{u}t + 4.0t^{2} \; \hat{i} + t^{2} \; \hat{j}\vec{r}$$

$$= \left (10.0 \; \hat{j} \right)t + 4.0t^{2} \; \hat{i} + t^{2} \; \hat{j}\times \;\hat{i} + y \;\hat{j}$$

$$= 4.0t^{2} \; \hat{i} + \left ( 10.0 \; t + t^{2} \right )\; \hat{j}$$

Since the motion of the particle is confined to the x-y plane, on equating the coefficients of

$$\hat{i} \; and \; \hat{j}$$, we get:

$$x = 4t^{2}$$

t =$$\left (\dfrac{x}{4} \right)^{\dfrac{1}{2}}$$

y = $$10 t + t^{2}$$

When x = 16 m:

$$t = \left (\dfrac{16}{4} \right )^{\dfrac{1}{2}}$$ = 2 s

Therefore, y = 10 × 2 + $$\left (2 \right)^{2}$$ = 24 m

(b) Velocity of the particle:

$$\vec{v} ( t ) = 8.0 t \; \hat{i} + 2.0 t \; \hat{j} + \vec{u}$$

At t = 2 s:

$$\vec{v} ( t ) = 8.0 \times 2 \; \hat{i} + 2.0 \times 2 \; \hat{j} + 20 \; \hat{j}\vec{v} ( t ) = 16 \; \hat{i} + 14\; \hat{j}$$

Therefore, Speed of the particle:

$$\left |\vec{v} \right | = \sqrt{\left ( 16 \right )^{2} + \left ( 14 \right )^{2}}\left |\vec{v} \right | = \sqrt{256 + 196}\left |\vec{v} \right | = \sqrt{452}\left |\vec{v} \right | = 21.26 \; m \; s^{-1}$$

Answered by Pragya Singh | 1 year ago

### Related Questions

#### Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by

(a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by

$$\theta (t)=tan^{-1}(\dfrac{v_{0y-gt}}{v_{ox}})$$

(b) Shows that the projection angle θ0 for a projectile launched from the origin is given by

$$\theta_{0}=tan^{-1}(\dfrac{4h_{m}}{R})$$

where the symbols have their usual meaning

#### A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of a radius of 80 m

A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of a radius of 80 m, he applies brakes and reduces his speed at the constant rate of 0.50 m/s every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn?

#### A fighter plane flying horizontally at an altitude of 1.5 km with a speed of 720 km/h passes directly

A fighter plane flying horizontally at an altitude of 1.5 km with a speed of 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s-1 to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take g = 10 m s-2 ).

#### A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away.

A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to be fixed, and neglect air resistance.

#### Can we associate a vector with (i) a sphere

Can we associate a vector with

(i) a sphere

(ii) the length of a wire bent into a loop

(iii) a plane area

Clarify for the same.