M-61Integration 8 : Kinematics

Note that some problems involve differentiation as well.

In all problems:

  • s represents distance from the origin in cm
  • v represents the velocity in cms-1
  • a represents the acceleration cms-2
  • t represents the time in seconds.

1.A particle starts motion 12cm from the origin at time t = 0. It moves in such a way that its velocity v at time t is given by the formula:

v = 4t – 2

(a)Find an expression for its distance, s, from the origin at time t.

(b)Show that the particle will never reach the origin.

(c) How far from the origin is the particle at the time when it is momentarily at rest?

(d) What is the acceleration of the particle?

2.The velocity of a particle, v, at time t is given by the formula

v =

The particle starts from the origin at time t = 0.

(a) Find an expression for the distance, s, of the particle from the origin at time t.

(b) Find the time at which the particle first returns to the origin.

(c)Find (i) the velocity, and (ii) the acceleration of the particle at the time referred to in (b).

3.The following facts are known about the motion of a particle:

(i)a = 1.5t – 2

(ii)when t = 4, v = 2

(iii)when t = 2, s = 0

Find expressions for v and s in terms of t.

4.A particle moves in such a way that its acceleration at time t is given by the formula

It starts at rest 20cm from the origin.

(a) Find an expression for the velocity in terms of t.

(b) What is the limit of the velocity as time tends to infinity? (As t gets very big, what number does v get closer and closer to?)

(c) Show that the distance from the origin at time t is given by the formula

Give answers correct to 3 significant figures below:

(d) At what time is the acceleration 4 cms-2 ?

(e)At the time in (c), how far is the particle from the origin?

(f)When will the velocity be 5 cms-1 ?

5.A particle moves so that its velocity at time t is given by

It is also know that at t = 0 the particle is 10cm from the origin.

(a)Find the distance of the particle from the origin at the time when its acceleration is zero.

(b)Find the distance of the particle from the origin at the time (t > 0) when it is momentarily at rest.