Chapter 2: Q20P (page 33)

(a) If the position of a particle is given by $x = 20 t - 5 t^{2}$ , where x is in m and t in secwhen, if ever, the particle’s velocity is zero? (b) When is its acceleration a zero? (c) For what time range (positive or negative) is a negative? (d) Positive? (e) Graph x(t), v(t) and a(t).

Short Answer

Expert verified

(a) Particle’s velocity is zero at $t = 1.2 s$

(b) Particle’s acceleration is zero at $t = 0 s$

(d) Range is positive at $t < 0$

(e) Graph of $x (t)$ , $v (t)$ and $a (t)$

Step by step solution

Given information

$x (t) = 20 t - 5 t^{3}$

To understand the concept of simple mathematical operation

This problem involves simple mathematical operation of derivative. Here differentiating the given equation, the equation for velocity and acceleration can be found. Further, by pluggingand, time whereandare zero can be calculated.

Formula:

$v (t) = \frac{d x (t)}{d t}$

Derive the expressions for velocity and acceleration

$x (t) = 20 t - 5 t^{2}$

Expressions for velocity and acceleration can be found by taking derivative of above equation with respect to t,

$v (t) = \frac{d x (t)}{d t} = \frac{d (20 t - 5 t^{2})}{d t} = 20 - 15 t^{2} a (t) = \frac{d v (t)}{d t} = \frac{d (20 - 15 t^{2})}{d t} = - 30 t$

(a) Calculate the time at which velocity is zero

$v (t) = 20 - 15 t^{2} 0 = 20 - 15 t^{2}$

$t = \sqrt{\frac{20}{15}} = 1.2 s$

(b) Calculate the time at which acceleration is zero

$a (t) = - 30 t 0 = - 30 t t = 0 s$

(c)Calculate the time range in which acceleration is negative

From equation, if acceleration $a (t) = - 30 t$ at $t > 0$ acceleration is negative.

(d) Calculate the time range in which acceleration is positive

From equation, if acceleration $a (t) = - 30 t$ at $t < 0$ acceleration is positive.

(e) Graph x(t), v(t) and a(t)

Graphs are shown below:

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(a) If the position of a particle is given by $x = 20 t - 5 t^{2}$ , where x is in m and t in secwhen, if ever, the particle’s velocity is zero? (b) When is its acceleration a zero? (c) For what time range (positive or negative) is a negative? (d) Positive? (e) Graph x(t), v(t) and a(t).

Short Answer

Step by step solution

Given information

To understand the concept of simple mathematical operation

Derive the expressions for velocity and acceleration

(a) Calculate the time at which velocity is zero

(b) Calculate the time at which acceleration is zero

(c)Calculate the time range in which acceleration is negative

(d) Calculate the time range in which acceleration is positive

(e) Graph x(t), v(t) and a(t)

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(a) If the position of a particle is given byx=20t-5t2, where x is in m and t in secwhen, if ever, the particle’s velocity is zero? (b) When is its acceleration a zero? (c) For what time range (positive or negative) is a negative? (d) Positive? (e) Graph x(t), v(t) and a(t).

Short Answer

Step by step solution

Given information

To understand the concept of simple mathematical operation

Derive the expressions for velocity and acceleration

(a) Calculate the time at which velocity is zero

(b) Calculate the time at which acceleration is zero

(c)Calculate the time range in which acceleration is negative

(d) Calculate the time range in which acceleration is positive

(e) Graph x(t), v(t) and a(t)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Turning Points in Physics

Measurements

Rotational Dynamics

Fundamentals of Physics

Modern Physics

Study anywhere. Anytime. Across all devices.

(a) If the position of a particle is given by $x = 20 t - 5 t^{2}$ , where x is in m and t in secwhen, if ever, the particle’s velocity is zero? (b) When is its acceleration a zero? (c) For what time range (positive or negative) is a negative? (d) Positive? (e) Graph x(t), v(t) and a(t).