Differential equations

Show that while the force is acting the motion of P is modelled by the differential equation

Hence find v in terms of t, showing that this expression for v is valid for 0 ⩽ t ⩽ π.

Determine the total distance travelled by P in the interval 0 ⩽ t ⩽ 10. Give your answer in an

exact form.

Show that B’s motion can be modelled by the differential equation =-4.

Solve the differential equation in part (a) to find the particular solution for v in terms of x

and u.

By considering the behaviour of v as x → ∞ describe one feature of the model that is not

realistic.

At the instant when B reaches the point A, where x = X, its speed is V ms^-1. The work done by the resistance as B moves from O to A is denoted by W J.

(c)

Explain the relevance of the sign of your answer in part (c)(i).

By writing your answer to part (c)(i) in terms of V and u show how the quantity W

relates to the energy of B.

A particle P of mass 2 kg moves along the x-axis.

At time t = 0, P passes through the origin O with speed 3 ms^-1.

At time t seconds the displacement of P from O is x m and the velocity of P is v ms^-1, where t ⩾ 0, x ⩾ 0 and v ⩾ 0.

While P is in motion the only force acting on P is a resistive force F of magnitude (v² + 1) N acting in the negative x-direction.

Find an expression for v in terms of x.

Determine the distance travelled by P while its speed drops from 3 ms^-1 to 2 ms^-1.

Particle Q is identical to particle P. At a different time, Q is moving along the x-axis under the influence of a single constant resistive force of magnitude 1 N. When t ′ = 0, Q is at the origin and its speed is 3 ms^-1.

By comparing the motion of P with the motion of Q explain why P must come to rest at some

finite time when t < 6 with x < 9.

Sketch the velocity-time graph for P. You do not need to indicate any values on your sketch.

Determine the maximum displacement of P from O during P ’s motion.

A particle P of mass 2.5 kg is free to move along the x ‑axis. When its displacement from the origin is x m its velocity is v m s^-1.

At time t = 0 seconds, P is at the point where x = 1 and is travelling in the negative x ‑direction with speed 5 m s^-1.

At this time an impulse of I Ns is applied to P in the positive x ‑direction so that P moves in the positive x‑direction with speed 18 m s^-1.

Find the value of I.

Subsequently, whenever P is in motion, two forces act on it. The first force acts in the positive x‑direction and has magnitude N. The second force acts in the negative x‑direction and has magnitude 60 v N.

Show that the motion of P can be modelled by the differential equation = + b where a

and b are constants whose values should be determined.

By solving the differential equation derived in part (b) find an expression for v in terms of x.

Describe in detail the motion of P when t ⩾ 0.

The resistive force, F, on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, r, the velocity of the sphere, v, and the viscosity of the fluid, η. You are given that η is measured in Nm^-2s.

By considering its units, find the dimensions of viscosity.

A model of the resistive force suggests the following relationship: F = 6πη^αr^βv^γ.

Explain whether or not it is possible to use dimensional analysis to verify that the constant 6 π

is correct.

Use dimensional analysis to find the values of α, β and γ.

A sphere of radius r and mass m falls vertically from rest through the fluid. After a time t its velocity is v.

By setting up and solving a differential equation, show that e^-kt = where k =

As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.

Find, in terms of g and k, the terminal velocity of the sphere.

In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.

Describe how, according to the model, the terminal velocity of the sphere changes as the

viscosity of the fluid through which it falls increases.