8. Exercises

The acceleration of a jet-ski is (\displaystyle 0.9\mathbf{i}+0.7\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-2}}. Its initial velocity is 4\displaystyle \mathbf{i} \displaystyle \text{m}{{\text{s}}^{-1}} and its initial position is (\displaystyle 400\mathbf{i}+350\mathbf{j}) m. The unit vectors are directed east and north respectively.

a) Find the velocity of the jet-ski when t = 10 seconds.

b) Find the position of the jet-ski when t = 10 seconds.

A particle is set into motion on a smooth inclined plane. The initial velocity of the particle is 5\displaystyle \mathbf{i} \displaystyle \text{m}{{\text{s}}^{-1}}, its acceleration is -4\displaystyle \mathbf{j} \displaystyle \text{m}{{\text{s}}^{-2}} and its initial position is (\displaystyle 18\mathbf{i}+14\mathbf{j}) m, where \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular unit vectors that lie in the plane in which the particle is moving.

a) Find the velocity of the particle after it has been moving for 4 seconds.

b) Find the position of the particle after it has been moving for 8 seconds.

The acceleration of a body is (\displaystyle 2\mathbf{i}+3\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-2}}. If the body starts at rest at the origin and accelerates for 4 seconds find the velocity and position of the body at the end of the 4 seconds. The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular.

A particle has a constant acceleration (\displaystyle 0.2\mathbf{i}+0.3\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-2}}. Initially it has a velocity (\displaystyle 4\mathbf{i}– 8\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}} and is at the point with position vector (\displaystyle 6\mathbf{i}+2\mathbf{j}) m.

a) Find the speed of the particle after 10 seconds.

b) Find the position vector of the particle after 10 seconds.

c) When is the velocity of the particle (\displaystyle 4.8\mathbf{i}– 6.8\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}}?