Calculus in kinematics

In this question you must show detailed reasoning.

A particle moves in a straight line. Its velocity t seconds after leaving a fixed point on the line is v m s^-1 where v = 2 + 3t + 0.6t².

Find the distance travelled by the particle from time t = 0 until the instant when its acceleration is

15 m s^-2.

A particle P moves in a straight line. The velocity v ms^-1 of P at time t seconds is given by

where k is a positive constant.

Sketch, on the axes in the Printed Answer Booklet, the velocity-time graph for P for values of

t from 0 to 9.

State the value of t in the interval 0 ⩽ t ⩽ 9 when the acceleration of P is zero.

In this question you must show detailed reasoning.

You are given that the total distance travelled by P in the interval 0 ⩽ t ⩽ 9 is 84 m.

Find the value of k.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle P is moving on a smooth horizontal surface under the action of a single force F N. At time t seconds, where t ⩾ 0, the velocity v ms^-1 of P, relative to a fixed origin O, is given by

v = (1 - 2t)i + (2t² + t - 13)j.

Show that P is never stationary.

Find, in terms of i and j, the acceleration of P at time t.

The mass of P is 0.5 kg.

Determine the magnitude of F when P is moving in the direction of the vector - 2 i + j. Give

your answer correct to 3 significant figures.

When t = 1, P is at the point with position vector .

Determine the bearing of P from O at time t = 1.5.

In this question the unit vectors i and j are in the directions east and north respectively.

At time t seconds, where t ⩾ 0, a particle P of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force (- 8 i - 54 j)N and a variable horizontal force (4ti + 6(2t - 1)²j)N.

Determine the value of t when the forces acting on P are in equilibrium.

It is given that P is at rest when t = 0.

Determine the speed of P at the instant when P is moving due north.

Determine the distance between the positions of P when t = 0 and t = 3.

A car is travelling on a straight horizontal road. The velocity of the car, vms^-1, at time t seconds as it travels past three points, P, Q and R, is modelled by the equation

v = at² + bt + c,

where a, b and c are constants.

The car passes P at time t = 0 with velocity 8ms^-1.

State the value of c.

The car passes Q at time t = 5 and at that instant its deceleration is 0.12 ms^-2. The car passes R at time t = 18 with velocity 2.96 ms^-1.

Determine the values of a and b.

Find, to the nearest metre, the distance between points P and R.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle R of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force F N. At time t seconds, the velocity v ms^-1 of R, relative to a fixed origin O, is given by v = (pt² - 3t)i + (8t + q)j, where p and q are constants and p < 0.

Given that when t = 0.5 the magnitude of F is 20, find the value of p.

When t = 0, R is at the point with position vector (2 i - 3 j) m.

Find, in terms of q, an expression for the displacement vector of R at time t.

When t = 1, R is at a point on the line L, where L passes through O and the point with position vector 2 i - 8 j.

Find the value of q.

The velocity v m s^-1 of a car at time t s, during the first 20 s of its journey, is given by v = kt + 0.03t², where k is a constant. When t = 20 the acceleration of the car is 1.3 m s^-2. For t > 20 the car continues its journey with constant acceleration 1.3 m s^-2 until its speed reaches 25 m s^-1.

Find the value of k.

Find the total distance the car has travelled when its speed reaches 25 m s^-1.