Motion in a Plane

A particle moves in a plane such that its position vector at time t is given by \( \vec{r} = (3t)\hat{i} + (4t)\hat{j} \). What is the magnitude of its velocity?

A particle moves in a circle of radius 10 m with a constant speed of 5 m/s. What is the magnitude of its acceleration?

A projectile is launched at an angle of 45° with initial speed 20 m/s. What is the horizontal range? (Take \( g = 9.8 \) m/s²)

If the position vector of a particle is given by \( \vec{r} = (t^2)\hat{i} + (t^3)\hat{j} \), what is the acceleration vector?

Two vectors \( \vec{A} = 3\hat{i} + 4\hat{j} \) and \( \vec{B} = 4\hat{i} - 3\hat{j} \) are added. What is the magnitude of \( \vec{A} + \vec{B} \)?

A particle moves with velocity \( \vec{v} = 5\hat{i} - 12\hat{j} \) m/s. What is the direction of velocity vector with respect to the positive x-axis?

A particle moves in a plane such that its acceleration vector is always perpendicular to its velocity vector. Which of the following motions is possible?

A particle moves such that its position vector is \( \vec{r} = (4t - t^2)\hat{i} + (3t^2)\hat{j} \). What is the time when velocity vector is perpendicular to acceleration vector?

A particle moves in a plane with position vector \( \vec{r} = (3\cos t)\hat{i} + (4\sin t)\hat{j} \). What is the magnitude of velocity at any time t?

A particle moves such that its velocity vector is \( \vec{v} = (2t)\hat{i} + (3)\hat{j} \). What is the acceleration vector?