Q8E The motion of a set of particles... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 1: Q8E (page 1)

The motion of a set of particles moving along the x鈥慳xis is governed by the differential equation $\frac{dx}{dt} = t^{3} - x^{3},$ where $x (t)$ denotes the position at time t of the particle.
猞� If a particle is located at $x = 1$ when $t = 1$ , what is its velocity at this time?
猞� Show that the acceleration of a particle is given by $\frac{d^{2} x}{{dt}^{2}} = 3 t^{2} - 3 t^{3} x^{2} + 3 x^{5} .$
猞� If a particle is located at $x = 2$ when $t = 2.5$ , can it reach the location $x = 1$ at any later time?
[Hint: $t^{3} - x^{3} = (t - x) (t^{2} + xt + x^{2}) .$ ]

Short Answer

Expert verified

猞� 7

猞� $\frac{d^{2} x}{{dt}^{2}} = 3 t^{2} - 3 t^{3} x^{2} + 3 x^{5}$

猞� No

Step by step solution

1(a): Finding the velocity by putting x=1 and t=2 in dxdt

Given, $\frac{dx}{dt} = t^{3} - x^{3},$

Substituting $x = 1$ and $t = 2$ , one gets,

$\frac{dx}{dt} = 2^{3} - 1^{3} = 8 - 1 = 7$

i.e., Velocity at time $t = 2$ and position $x = 1$ is 7.

Hence, the velocity is 7.

2(b): Differentiating dxdt with respect to t.

$\frac{dx}{dt}$ represents velocity while $\frac{d^{2} x}{{dt}^{2}}$ gives the acceleration of the particle.

$\frac{dx}{dt} = t^{3} - x^{3}, \frac{d^{2} x}{{dt}^{2}} = 3 t^{2} - 3 x^{2} . \frac{dx}{dt} = 3 t^{2} - 3 x^{2} (t^{3} - x^{3}) = 3 t^{2} - 3 x^{2} t^{3} + 3 x^{5}$

Hence, it is shown that the acceleration of a particle is given by $\frac{d^{2} x}{{dt}^{2}} = 3 t^{2} - 3 t^{3} x^{2} + 3 x^{5} .$

3(c): Analyzing the position of particle at different time

From the graph, it is visible that the particle is stabilized about the velocity $\frac{dx}{dt} = 1$

Now, if $x = 1$ and $t = 2.5$ , then $\frac{dx}{dt} > 0$ , i.e., the particle moves away from $x = 1$ .

Thus, the position of the particle keeps on increasing and can reach x = 1 only once.

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Short Answer

Step by step solution

1(a): Finding the velocity by putting x=1 and t=2 in dxdt

2(b): Differentiating dxdt with respect to t.

3(c): Analyzing the position of particle at different time

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