Scalar product

In this question you must show detailed reasoning.

A particle P of mass 3.25 kg is moving on a smooth horizontal plane with velocity v ms^-1 where v = 4 i - 6 j.

Find the kinetic energy of P.

A system of horizontal forces starts to act on P. One of the forces is F N where F = i - (3 + t) j and t is the time, in seconds, after the system of forces starts to act.

Find the power developed by F at time t = 0.

Given that the velocity of P does not change at any time, find the total power developed by

the forces other than F at time t = 6.

The points P, Q and R have coordinates (0, 2, 3), (2, 0, 1) and (1, 3, 0) respectively.

The acute angle between the line segments PQ and PR is θ.

The triangle PQR lies in the plane Π.

Determine an equation for Π, giving your answer in the form ax + by + cz = d, where a, b, c

and d are integers.

A particle P of mass 3 kg is moving on a smooth horizontal surface under the influence of a variable horizontal force F N. At time t seconds, where t ⩾ 0, the velocity of P, v ms^-1, is given by

v = (32 sinh (2 t)) i + (32 cosh (2 t) - 257) j.

By considering kinetic energy, determine the work done by F over the interval

0 ⩽ t ⩽ ln 2.

Explain the significance of the sign of the answer to part (a)(i).

Determine the rate at which F is working at the instant when P is moving parallel to the

i-direction.

The coordinates of two points, A and B, are (- 1, 6) and (5, 12) respectively, where the units of the coordinate axes are metres. A particle P moves from A to B under the action of several forces. The force F = 7 i - 2 j N is one of the forces acting on P.

Calculate the work done by F on P as P moves from A to B.

At the instant when P reaches B its velocity is -i - 5j m s^-1.

Find the power generated by F at the instant that P reaches B.

One end of a light elastic string was attached to the origin of the coordinate system and the other to P when P was at A, before it moved to B. The natural length of the string is 8 m and its modulus of elasticity is 24 N.

At the instant that P reaches B, find the following.

• The tension in the string

• The elastic potential energy stored in the string

Find the position vector of the point of intersection of l₁ and Π.

Find the acute angle between l₁ and Π.

A is the point on l₁ where λ = 1.

l₂ is the line with the following properties.

• l₂ passes through A

• l₂ is perpendicular to l₁

• l₂ is parallel to Π

Find, in vector form, the equation of l₂.

Calculate the work done by the force during these 5 seconds.

Calculate the average power generated by the force during these 5 seconds.

The speed of P at O is 10ms^-1.

Calculate the speed of P at A.