Chapter 2: Q. 36 (page 61)

The position of a particle is given by the function $x = 2 t^{3} - 6 t^{2} + 12$ m, where t is in s. a. At what time does the particle reach its minimum velocity? What is ${(V_{x})}_{m i n}$ ? b. At what time is the acceleration zero?

Short Answer

Expert verified

(a)The particle reaches its minimum velocity at $t = 1 s$ . The minimum velocity of the particle is $- 6 \frac{m}{s}$ .

(b) The acceleration of the particle is zero at $t = 1 s$ .

Step by step solution

Given information

The position of the particle is given by $x = 2 t^{3} - 6 t^{2} + 12$
.

The velocity of the particle.

The velocity is given by $v_{x} = \frac{d x}{d t}$ .

Differentiate the given function with respect to $t$ .

$v_{x} = \frac{d}{d t} (2 t^{3} - 6 t^{2} + 12) = 6 t^{2} - 12 t$

Acceleration of the particle.

To find the acceleration, differentiate $v_{x} = 6 t^{2} - 12 t$ with respect to $t$ .

$a_{x} = \frac{d}{d t} (6 t^{2} - 12 t) = 12 t - 12$

Substitute 0 for $a_{x}$ .

$0 = 12 t - 12 12 t = 12 t = 1 s$

Part (a)

The velocity of the particle at $t = 1 s$ is

$v_{x} = 6 {(1)}^{2} - 12 (1) = 6 - 12 = - 6 \frac{m}{s}$

The particle reaches its minimum velocity at $t = 1 s$ . The minimum velocity of the particle is $- 6 \frac{m}{s}$ .

Part (b)

The acceleration of the particle is zero at $t = 1 s$ .

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The position of a particle is given by the function $x = 2 t^{3} - 6 t^{2} + 12$ m, where t is in s. a. At what time does the particle reach its minimum velocity? What is ${(V_{x})}_{m i n}$ ? b. At what time is the acceleration zero?

Short Answer

Step by step solution

Given information

The velocity of the particle.

Acceleration of the particle.

Part (a)

Part (b)

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The position of a particle is given by the function x=2t3-6t2+12m, where t is in s. a. At what time does the particle reach its minimum velocity? What is Vxmin? b. At what time is the acceleration zero?

Short Answer

Step by step solution

Given information

The velocity of the particle.

Acceleration of the particle.

Part (a)

Part (b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Circular Motion and Gravitation

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Classical Mechanics

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

The position of a particle is given by the function $x = 2 t^{3} - 6 t^{2} + 12$ m, where t is in s. a. At what time does the particle reach its minimum velocity? What is ${(V_{x})}_{m i n}$ ? b. At what time is the acceleration zero?