8. Exercises

The acceleration of a jet-ski is (\displaystyle 0\textrm{.}9\mathbf{i}+0\textrm{.}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\textrm{.}2\mathbf{i}+0\textrm{.}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\textrm{.}8\mathbf{i}– 6\textrm{.}8\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}}?

A light aeroplane has a velocity of 80\displaystyle \mathbf{i} \displaystyle \text{m}{{\text{s}}^{-1}}, as it is moving along a runway. When it takes off it then experiences an acceleration of (\displaystyle \mathbf{i}+4\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-2}} for the first 30 seconds of its flight. The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are directed horizontally and vertically respectively. Assume that the aeroplane is at the origin when it takes off and begins to accelerate.

a) Find expressions for the velocity and position of the aeroplane at time t seconds.

b) Find the speed of the aeroplane when it is at a height of 50 m.

A boat has initial velocity (\displaystyle \mathbf{i}+2\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}}. It moves with a constant acceleration, \displaystyle \mathbf{a} until after 10 seconds its velocity is (\displaystyle 6\mathbf{i}-8\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}}. The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are directed east and north respectively.

(a) Find \displaystyle \mathbf{a} .

(b) The boat then travels with this velocity for a further 40 seconds. Find the distance between the initial and final positions of the boat.

A football is kicked so that its initial velocity is (\displaystyle 8\mathbf{i}+10\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}}. Assume that its acceleration is -10\displaystyle \mathbf{j} \displaystyle \text{m}{{\text{s}}^{-2}}, where the unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are horizontal and vertical unit vectors respectively. Assume that the ball is kicked from the origin, which is on a horizontal surface.

a) After the ball has been moving for T seconds it hits the ground for the first time. Find T.

b) Find the distance of the point where the ball hits the ground for the first time from the origin.

c) Find the maximum height of ball.

The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are perpendicular and lie in a horizontal plane. A particle moves from the origin. Its initial velocity was (\displaystyle 4\mathbf{i}+6\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}} and after 20 seconds its velocity is (\displaystyle 24\mathbf{i}+46\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}}. The acceleration of the particle is constant.

a) Find the acceleration of the particle.

b) Find the distance of the particle from the origin after accelerating for 30 seconds.

The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are horizontal and vertical respectively. An arrow is shot so that it starts to move with velocity (\displaystyle 15\mathbf{i}+18\mathbf{j}) \displaystyle \text{m}{{\text{s}}^{-1}} from the point with position vector 1.5\displaystyle \mathbf{j} m. After 3 seconds it hits a target. Assume that the acceleration of the particle is -10\displaystyle \mathbf{j} \displaystyle \text{m}{{\text{s}}^{-2}} and that the arrow can be modelled as a particle.

a) Find the position of the point where the arrow hits the target.

b) Find the speed of the arrow when it hits the target.

A radio controlled aeroplane has initial position 10\displaystyle \mathbf{k} m and at this time is flying north east with a speed of \displaystyle 8\sqrt{2} \displaystyle \text{m}{{\text{s}}^{-1}}. It has an acceleration of \displaystyle (0\textrm{.}01\mathbf{i}+0\textrm{.}02\mathbf{j}+0\textrm{.}1\mathbf{k}) \displaystyle \text{m}{{\text{s}}^{-2}}. The unit vectors \displaystyle \mathbf{i}, \displaystyle \mathbf{j} and \displaystyle \mathbf{k} are directed east, north and vertically upwards respectively. The aeroplane stops accelerating when it reaches a height of 90 metres.

a) Find the time for which the aeroplane is accelerating.

b) Find the speed of the aeroplane when it stops accelerating.

c) A car drives so that it is always directly below the aeroplane. Find the distance between the initial position of the car and its position when the aeroplane stops accelerating.