Motion in a Plane

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

A particle moves along a circular path 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 the horizontal from the ground. What is the ratio of horizontal range to maximum height?

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

A boat crosses a river 200 m wide flowing at 3 m/s. The boat's speed in still water is 4 m/s and it is aimed directly across the river. What is the time taken to cross?

The position vector of a particle moving in a plane is given by \( \vec{r}(t) = (t^2)\hat{i} + (2t)\hat{j} \). What is the acceleration vector?

A particle moves in a plane with velocity \( \vec{v} = (3t)\hat{i} + (4)\hat{j} \) m/s. What is the magnitude of acceleration at time \( t=2 \) s?

A particle moves such that its position vector is given by \( \vec{r}(t) = (5\cos t)\hat{i} + (5\sin t)\hat{j} \). What is the speed of the particle?

Two vectors \( \vec{A} = 3\hat{i} + 4\hat{j} \) and \( \vec{B} = 4\hat{i} - 3\hat{j} \) are given. What is the angle between them?

A particle moves in a plane with velocity \( \vec{v} = (4t)\hat{i} + (3)\hat{j} \). What is the magnitude of acceleration?