Calculus in kinematics

[In this question, position vectors are given relative to a fixed origin O.] At time t seconds, where t 0, the position vector of a particle P is r metres where

Find the position vector of P at t = 4

Find the exact distance of P from O at t = 4

Find an expression for the velocity of P at time t seconds, where t 0, giving your answer in terms of t, i and j At t = T, the acceleration of P is in a direction that is perpendicular to the line with

Find the value of T.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. [ In this question, i is a unit vector due east and j is a unit vector due north. Position vectors are given relative to a fixed origin O. ] At time t seconds, t 1, the position vector of a particle P is r metres, where

and c is a constant. Whent = 4, the bearing of P from O is 135°

Show that c = 3

Find the speed of P whent = 4 Whent = T, P is accelerating in the direction of (–i – 27j).

Find the value of T.

At time t seconds, where t 0, a particle P has velocity vm s^–1 where

Find

the speed of P at time t = 0

the value of t when P is moving parallel to (i + j)

the acceleration of P at time t seconds

the value of t when the direction of the acceleration of P is perpendicular to i

At time t seconds, a particle P has velocity vm s⁻¹, where

Find the acceleration of P at time t seconds, where t 0

Find the value of t at the instant when P is moving in the direction of i − j

At time t seconds, where t 0, the position vector of P, relative to a fixed origin O, is r metres.

When t = 1, r = −j

Find an expression for r in terms of t.

Find the exact distance of P from O at the instant when P is moving with

TOTAL FOR MECHANICS IS 50 MARKS

At time t seconds, a particle P has velocity vm s⁻¹, where

Find the acceleration of P at time t seconds, where t 0

Find the value of t at the instant when P is moving in the direction of i − j At time t seconds, where t 0, the position vector of P, relative to a fixed origin O, is r metres. When t = 1, r = −j

Find an expression for r in terms of t.

Find the exact distance of P from O at the instant when P is moving with

TOTAL FOR MECHANICS IS 50 MARKS

At time t seconds, where t 0, a particle P moves so that its acceleration am s⁻² is given by

At the instant when t = 0, the velocity of P is 36im s⁻¹

Find the velocity of P when t = 4

Find the value of t at the instant when P is moving in a direction perpendicular to i

At time t seconds, where t 0, a particle Q moves so that its position vector r metres, relative to a fixed origin O, is given by

Find the value of t at the instant when the speed of Q is 5 m s⁻¹

A particle P of mass 0.5 kg moves under the action of a single force F newtons. At time t seconds, t 0, P has velocity v m s^–1, where

Find

the magnitude of F when t = 3

the acceleration of P at the instant when it is moving in the direction of the

When t = 1, P is at the point A. When t = 2, P is at the point B.

Find, in terms of i and j, the vector AB

At time t = 0 a particle P leaves the origin O and moves along the x-axis. At time t seconds, the velocity of P is v m s^–1 in the positive x direction, where

The particle is instantaneously at rest when t = t₁ and when t = t₂ (t₁ t₂).

Find the value of t₁ and the value of t₂.

Find the magnitude of the acceleration of P at the instant when t = t₁.

Find the distance travelled by P in the interval t₁ ≤ t ≤ t₂.

Show that P does not return to O.

A particle P moves along a straight line. The speed of P at time t seconds (t 0) is v m s^–1, where v = (pt² + qt + r) and p, q and r are constants. When t = 2 the speed of P has its minimum value. When t = 0, v = 11 and when t = 2, v = 3 Find

the acceleration of P when t = 3

the distance travelled by P in the third second of the motion.

At time t seconds, where t 0, a particle P is moving on a horizontal plane with acceleration [(3t² – 4t)i + (6t – 5)j] m s^–2. When t = 3 the velocity of P is (11i + 10j) m s^–1. Find

the velocity of P at time t seconds,

the speed of P when it is moving parallel to the vector i.

A particle P moves on the x-axis. At time t seconds the velocity of P is v m s^–1 in the direction of x increasing, where

Find

the times when P is instantaneously at rest,

the greatest speed of P in the interval 0 ≤ t 4

the total distance travelled by P in the interval 0 t ≤ 4

A particle P moves along a straight line in such a way that at time t seconds its velocity v m s^–1 is given by

Find

the times when P is at rest,

the total distance travelled by P between t = 0 and t = 4.

At time t seconds the velocity of a particle P is [(4t – 5)i + 3j] m s^–1. When t = 0, the position vector of P is (2 i + 5 j) m, relative to a fixed origin O.

Find the value of t when the velocity of P is parallel to the vector j.

Find an expression for the position vector of P at time t seconds.

A second particle Q moves with constant velocity (−2i + cj) m s⁻¹. When t = 0, the position vector of Q is (11 i + 2 j) m. The particles P and Q collide at the point with position vector (d i +14 m. j)

Find

the value of c,

the value of d.

[ In this question i and j are perpendicular unit vectors in a horizontal plane. ] A particle P moves in such a way that its velocity v m s^–1 at time t seconds is given by

Find the magnitude of the acceleration of P when t = 1

Given that, when t = 0, the position vector of P is i metres,

find the position vector of P when t = 3

A particle P moves on the x-axis. The acceleration of P at time t seconds is t 4 m s in ² the positive x-direction. The velocity of P at time t seconds is v m s^–1. When t v = 0, = 6. Find

v in terms of t,

the values of t when P is instantaneously at rest,

the distance between the two points at which P is instantaneously at rest.

A particle moves along the x-axis. At time t = 0 the particle passes through the origin with speed 8 m s^–1 in the positive x-direction. The acceleration of the particle at time t seconds, t 0, is (4t³ – 12t) m s^–2 in the positive x-direction. Find

the velocity of the particle at time t seconds,

the displacement of the particle from the origin at time t seconds,

the values of t at which the particle is instantaneously at rest.

A particle P moves on the x-axis. The acceleration of P at time t seconds, t 0, is (3t + 5) m s^–2 in the positive x-direction. When t = 0, the velocity of P is 2 m s^–1 in the positive x-direction. When t = T, the velocity of P is 6 m s^–1 in the positive x-direction. Find the value of T.

A particle P moves along the x-axis. At time t seconds the velocity of P is v m s^–1 in the positive x-direction, where v = 3t² – 4t + 3. When t = 0, P is at the origin O. Find the distance of P from O when P is moving with minimum velocity.

At time t = 0 a particle P leaves the origin O and moves along the x- axis. At time t seconds the velocity of P is v m s^–1, where

Find the maximum value of v.

Find the time taken for P to return to O.

A particle P of mass 0.5 kg is moving under the action of a single force F newtons. At time t seconds,

The velocity of P at time t seconds is v m s^–1. When t = 0, v = i – 4j.

Find v at time t seconds.

When t = 3, the particle P receives an impulse (–5 i + 12 j) N s.

Find the speed of P immediately after it receives the impulse.

At time t seconds (t 0), a particle P has position vector p metres, with respect to a fixed origin O, where

Find

the velocity of P at time t seconds,

the value of t when P is moving parallel to the vector i.

When t = 1, the particle P receives an impulse of (2 i – 6 j) N s. Given that the mass of P is 0.5 kg,

find the velocity of P immediately after the impulse.