Rate of Change Worksheet

**Page 1**

1.

The displacement (in meters) of a particle in time $t$ is given by $s$ = - $t$^{3} + 8$t$^{2} + 7$t$ + 2. Find the velocity of the particle when $t$ = 2 sec.

a. | 40 m/s | ||

b. | 27 m/s | ||

c. | 80 m/s | ||

d. | 28 m/s | ||

e. | 76 m/s |

[Write the function.]

[Find the velocity of the particle,

= - 3

[Differentiate with respect to

The velocity of the particle at

[Substitute

Therefore, the velocity of the particle at

Correct answer : (2)

2.

The distance travelled by a particle on a straight line in time $t$ is given by $s$ = 5 + 6$t$ - $t$^{2} + 7$t$^{3} where $s$ is in feet and $t$ is in seconds. Find the velocity of the particle when the acceleration is zero.

a. | 5 ft/sec | ||

b. | 6 ft/sec | ||

c. | $\frac{50}{7}$ ft/sec | ||

d. | $\frac{125}{21}$ ft/sec | ||

e. | $\frac{43}{7}$ ft/sec |

[Write the function.]

[Find the velocity of the particle,

= 6 - 2

[Differentiate with respect to

[Find the acceleration of the particle,

= - 2 + 42

[Differentiate with respect to

If the acceleration of the particle,

[Solve for

So, the acceleration of the particle is zero at

At

[Substitute

Therefore, the velocity of the particle when acceleration is zero is

Correct answer : (4)

3.

The displacement $s$ of a body with the time $t$ is related as $s$ = $\mathrm{ae}$^{3$t$} - $\mathrm{be}$^{- 4$t$}. What is the acceleration of the body at any time $t$?

a. | $a$ - $b$ | ||

b. | $\mathrm{ae}$ ^{3$t$} + $\mathrm{be}$^{- 4$t$} | ||

c. | 9$\mathrm{ae}$ ^{3$t$} - 16$\mathrm{be}$^{- 4$t$} | ||

d. | (3$a$ - 4$b$)(${e}^{3t}-{e}^{-4t}$) |

[Write the function.]

[Find the velocity of the body,

= 3

[Differentiate with respect to

= 3

[Find the acceleration of the body,

= 9

[Differentiate with respect to

= 9

Therefore, the acceleration of the body at any time

Correct answer : (3)

4.

A particle moves along a straight line with the relation between the time and distance ($s$) in such a way that $s$ = 9 - 5$t$^{2} + 10$t$^{3}. At what time will the acceleration be 230 units/sec^{2}?

a. | 300 sec | ||

b. | $\frac{1}{4}$ sec | ||

c. | 4 sec | ||

d. | $\frac{11}{3}$ sec | ||

e. | 280 sec |

[Find the velocity of the particle,

= - 10

[Use Difference, Sum Rule.]

[Find the acceleration of the particle,

= - 10 + 60

[Use Difference, Sum Rule.]

If the acceleration of the particle is 230 units/sec

[Solve for

Therefore, the acceleration of the particle is 230 units/sec

Correct answer : (3)

5.

A particle covers the displacement $s$ in time $t$ which is given as $s$ = $k$ ${t}^{\frac{1}{2}}$ where $k$ is a constant, then find the acceleration of the particle at $t$ = 4 sec.

a. | $\frac{k}{16}$ units/sec ^{2} | ||

b. | $k$ units/sec ^{2} | ||

c. | - $\frac{k}{32}$ units/sec ^{2} | ||

d. | $\frac{k}{32}$ units/sec ^{2} |

[Write the function.]

[Find the velocity of the particle,

=

[Differentiate with respect to

[Find the acceleration of the particle,

=

[Differentiate with respect to

The acceleration of the particle at

= -

Therefore, the acceleration of the particle at

Correct answer : (3)

6.

If the displacement $s$ of a body is related with time $t$ as $s$ = 2 cos w$t$ - 3 sin w$t$, then find the acceleration of the particle at time $t$.

a. | - $s$ | ||

b. | - ω ^{2} $s$ | ||

c. | ω [- 2 cos ω$t$ + 3 sin ω$t$] | ||

d. | - ω ^{2} [2 cos ω$t$ + 3 sin ω$t$] | ||

e. | ω ^{2} $s$ |

[Write the function.]

[Find the velocity of the body,

= - 2 (sin w

[Differentiate with respect to

= - w[2 sin w

[Factor.]

[Find the acceleration of the body,

= - w[2 (cos w

[Differentiate with respect to

= - w

[Factor.]

= - w

[Replace 2 cos w

Therfore, the acceleration of the body at time

Correct answer : (2)

7.

The displacement $s$ of a moving particle is related with time as $s$ = $\mathrm{at}$^{2} + $\mathrm{bt}$ + $c$. If the displacement after 1 sec is 20 m, the velocity after 2 sec is 30 m/sec and acceleration is 10 m/sec^{2}, then find the values of $a$, $b$ and $c$.

a. | 5, $\frac{146}{5}$and $\frac{-47}{5}$ | ||

b. | 5, 10 and 5 | ||

c. | 5, -10 and -5 | ||

d. | 20, -50 and -10 | ||

e. | 5, 50 and 5 |

[Write the function.]

At

[Substitute

[Find the velocity of the particle,

= 2

[Differentiate with respect to

The velocity of the particle after 2 sec is 30 m/sec.

2

[Substitute

Acceleration of the particle,

The acceleration of the particle after 2 sec is 10 m/sec

2

[Solve for

4(5) +

[Substitute

1(5) + 1(10) +

[Substitute

Therefore, the values of

Correct answer : (2)

8.

The displacement of a body moving on straight line in time $t$ is given as $s$ = 256$t$ - $\frac{16{t}^{3}}{3}$. At what instant will the velocity of the body vanishes?

a. | 256 sec | ||

b. | 384 sec | ||

c. | 4 sec | ||

d. | 16 sec | ||

e. | 19.59 sec |

[Write the function.]

[Find the velocity of the body,

= 256 -

[Differentiate with respect to

If the velocity of the body vanishes, then

256 - 16

[Solve for

At

[Time is non-negative.]

Correct answer : (3)

9.

A projectile is fired straight up from ground level with an initial velocity of 112 feet per second. Its height $s$ above the ground after time $t$ is given by $s$ = 112$t$ - 16$t$^{2}. What is its velocity and acceleration at $t$ = 2 sec?

a. | - 32 ft/sec, 48 ft/sec ^{2} | ||

b. | 96 ft/sec, - 16 ft/sec ^{2} | ||

c. | 48 ft/sec, - 32ft/sec ^{2} | ||

d. | 16 ft/sec, 32 ft/sec ^{2} | ||

e. | 80 ft/sec, - 16 ft/sec ^{2} |

[Write the function.]

[Find the velocity of the stone,

= 112 - 32

[Differentiate with respect to

The velocity of the stone at

[Substitute

[Find the acceleration of the stone,

= - 32

[Differentiate with respect to

The acceleration at

Therefore, after 2 seconds, the velocity, acceleration of the stone projected vertically upwards with an initial velocity of 112 ft/sec are 48 ft/sec and - 32 ft/sec

Correct answer : (3)

10.

The displacement (in feet) of the particle in time $t$ is given by $s$ = ln ($t$ + 3). What is the velocity of the particle at time $t$ = 2.6 sec?

a. | 1.722 ft/sec | ||

b. | 1.386 | ||

c. | 0.535 ft/sec | ||

d. | 0.748 ft/sec | ||

e. | 0.178 ft/sec |

[Write the function.]

[Find the velocity of the particle,

=

[Differentiate with respect to

The velocity of the particle at

[Substitute

[Simplify.]

The velocity of the particle at

Correct answer : (5)