Question

The position of a particle moving along the x axis depends on the time according to the equation x = ct²-bt³, where x is in meters and t in seconds. What are the units of (a) constant c and (b) constant b? Let their numerical values be 3.0 and 2.0, respectively. (c) At what time does the particle reach its maximum positive x position? From t=0.0s to t=4.0 s, (d) what distance does the particle move and (e) what is its displacement? Find its velocity at times (f) 1.0 s, (g) 2.0 s, (h) 3.0 s, and (i) 4.0 s. Find its acceleration at times (j) 1.0 s, (k) 2.0 s, (l) 3.0 s, and (m) 4.0 s.

Answer 1

The position of a particle moving along the x-axis depends on the time according to the equation -

x = c t² - b t³                                                                                    { eq.1 }

where, x = distance measured in meters (m)

t = time measured in seconds (s)

Differentiating eq.1 w.r.t time, we get

v = (dx / dt) = d [(c t² - b t³)] / dt

v = 2 c t - 3 b t²                                                                                   { eq.2 }

Again, differentiating eq.2 w.r.t time & we get

a = (dv / dt) = d [(2 c t - 3 b t²)] / dt

a = 2 c - 6 b t                                                                                       { eq.3 }

(a) What are the units of constant c?

using a dimensional analysis, we have

c t² = L $\Leftrightarrow$ c T² = L

c = L T^-2

which means that unit of constant 'c' will be m/s².

(b) What are the units of constant b?

using a dimensional analysis, we have

b t³ = L $\Leftrightarrow$ b T³ = L

b = L T^-3

which means that unit of constant 'b' will be m/s³.

(c) At what time does the particle reach its maximum positive x position?

using eq.2; v = 2 c t - 3 b t²

0 = t (2 c - 3 b t)

(3 b) t = (2 c)

t = (2 c) / (3 b) $\Rightarrow$ [(2 x 3 m/s²) / (3 x 2 m/s³)]

t = 1 sec

(d) The distance moved by a particle which will be given as -

using eq.1 ;   x = c t² - b t³

x = [(3 m/s²) (4 s)² - (2 m/s³) (4 s)³]

x = [(48 m) - (128 m)]

x = - 80 m

(e) The displacement of a particle from t = 0.0 s to t = 4.0 s which will be given as -

$\Delta$d = |x (t=4s) - x (t=0s) |

$\Delta$d = | [(3 m/s²) (4 s)² - (2 m/s³) (4 s)³] - [(3 m/s²) (0 s)² - (2 m/s³) (0 s)³] |

$\Delta$d = | [(48 m) - (128 m)] - [(0 m) - (0 m)] |

$\Delta$d = 80 m

(f) The velocity of a particle at t = 1 s which will be given as -

v = 2 c t - 3 b t²

v = [2 (3 m/s²) (1 s)] - [3 (2 m/s³) (1 s)²]

v = [(6 m/s) - (6 m/s)]

v = 0 m/s

(g) The velocity of a particle at t = 2 s which will be given as -

v = 2 c t - 3 b t²

v = [2 (3 m/s²) (2 s)] - [3 (2 m/s³) (2 s)²]

v = [(12 m/s) - (24 m/s)]

v = - 12 m/s

(h) The velocity of a particle at t = 3 s which will be given as -

v = 2 c t - 3 b t²

v = [2 (3 m/s²) (3 s)] - [3 (2 m/s³) (3 s)²]

v = [(18 m/s) - (54 m/s)]

v = - 36 m/s

(i) The velocity of a particle at t = 4 s which will be given as -

v = 2 c t - 3 b t²

v = [2 (3 m/s²) (4 s)] - [3 (2 m/s³) (4 s)²]

v = [(24 m/s) - (96 m/s)]

v = - 72 m/s

(j) An acceleration of a particle at t = 1 s which will be given as -

a = 2 c - 6 b t     

a = [2 (3 m/s²) - 6 (2 m/s³) (1 s)]

a = [(6 m/s²) - (12 m/s²)]

a = - 6 m/s²

(k) An acceleration of a particle at t = 2 s which will be given as -

a = 2 c - 6 b t     

a = [2 (3 m/s²) - 6 (2 m/s³) (2 s)]

a = [(6 m/s²) - (24 m/s²)]

a = - 18 m/s²

(l) An acceleration of a particle at t = 3 s which will be given as -

a = 2 c - 6 b t     

a = [2 (3 m/s²) - 6 (2 m/s³) (3 s)]

a = [(6 m/s²) - (36 m/s²)]

a = - 30 m/s²

(m) An acceleration of a particle at t = 4 s which will be given as -

a = 2 c - 6 b t     

a = [2 (3 m/s²) - 6 (2 m/s³) (4 s)]

a = [(6 m/s²) - (48 m/s²)]

a = - 42 m/s²