Position - Velocity - Acceleration Functions Worksheet

**Page 1**

1.

If the square of the displacement S traversed by a particle in time $t$ is given by S^{2} = $t$^{2} + 9$t$ + 64 , then choose the acceleration of the particle from the following.

a. | $\frac{175}{4{s}^{3}}$ | ||

b. | $\frac{175}{{s}^{2}}$ | ||

c. | $\frac{175}{4{s}^{4}}$ | ||

d. | $\frac{175}{{s}^{3}}$ |

[Take derivative of S.]

2S

SV =

[As

V =

[Take derivative of V.]

=

[Quotient rule of derivative.]

[Acceleration =

=

[Substitute V =

[Substitute

=

[Simplify.]

Correct answer : (1)

2.

The displacement function of a particle with time t in seconds is S = 4$t$^{3} - 9$t$^{2} + 4$t$ + 6 ft. Find the velocity of the particle when its acceleration becomes zero.

a. | 4 ft/s | ||

b. | 54 ft/s | ||

c. | 6 ft/s | ||

d. | $\frac{-11}{4}$ ft/s |

[Definition.]

The velocity = V =

=

[Substitute S from step1.]

ÞV = 12

[Simplify.]

The acceleration =

[Definition.]

=

[Substitute V from step 4.]

Þ

Suppose

So, the acceleration of the particle is zero, at

At

Correct answer : (4)

3.

The displacement S of a particle with respect to time $t$ is given by S = - 8$t$^{3} + 4$t$. The velocity of the particle

a. | Decreases with time | ||

b. | Is constant with respect to time | ||

c. | Increases and decreases with time | ||

d. | Increases with time |

The velocity of the particle V =

[Definition.]

=

[Substitute S from step1.]

Þ V = - 24

The acceleration of the particle =

[Definition.]

=

[Substitute V from step4.]

Þ

The acceleration is negative for

As acceleration is negative, with time

Correct answer : (1)

4.

The distance 'S' on a straight line traveled by a particle in time $t$ is given by S = 5$t$ - 2$t$^{3}. What is the maximum velocity of the particle?

a. | -1 | ||

b. | - 5 | ||

c. | 5 | ||

d. | 7 |

[The distance on a straight line is displacement.]

The velocity of the particle = V =

[Definitions.]

=

[Substitute S from step1.]

Þ V = 5 - 6

The maximum velocity of the particle = 5 - minimum value of 6

= 5 - 0

[Minimum value of

= 5

Correct answer : (3)

5.

The displacement 'S' described by a particle in $t$ seconds is given by S = 2$t$ - 2$t$^{2} ft . What is the velocity of the particle at any time $t$?

a. | (2 - 2$t$) ft/s | ||

b. | 2 ft/s | ||

c. | - 4 ft/s | ||

d. | (2 - 4$t$) ft/s |

Velocity of the particle at any time

[Definition.]

=

[Substitute S from step1.]

= (2 - 4

Correct answer : (4)

6.

The displacement S of a body in $t$ seconds is given by S = (4$t$^{3} - 7$t$^{2} + 4$t$ + 4) m. The instantaneous velocity of the body is

a. | (12$t$ ^{2} + 14$t$ + 4) m/s | ||

b. | (12$t$ ^{2} - 14$t$ + 4) m/s | ||

c. | (12$t$ ^{2} + 14$t$ - 4) m/s | ||

d. | ( 12$t$ ^{2} + 14$t$) m/s |

The instantaneous velocity of the particle =

[Definition.]

=

[Substitute S from step1.]

= (12

Correct answer : (2)

7.

The displacement of a body with time $t$ is related as S = $a$$e$^{$\mathrm{6t}$} - $b$$e$^{-4$t$}. Find the rate of change of displacement with respect to time $t$.

a. | 6$\mathrm{ae}$ ^{6$t$} + 4$\mathrm{be}$^{- 4$t$} | ||

b. | $\mathrm{ae}$ ^{$\mathrm{6t}$} - $\mathrm{be}$^{-4$t$} | ||

c. | $\mathrm{ae}$ ^{$\mathrm{6t}$} + $\mathrm{be}$^{-4$t$} | ||

d. | 36$\mathrm{ae}$ ^{$\mathrm{6t}$} - 16$\mathrm{be}$^{-4$t$} |

The rate of change of displacement with respect to time

[Definition.]

=

[Substitute S from step1.]

= 6

[Simplify.]

Correct answer : (1)

8.

The displacement S (in meters) of a particle is related with time $t$(in seconds) as S = 3$t$^{3} + 4$t$^{2} + 3$t$ + 2 The velocity of the particle when $t$ = 3 sec is

a. | 105 m/sec | ||

b. | 27 m/sec | ||

c. | 108 m/sec | ||

d. | 42 m/s |

Velocity of the particle = V =

[Definition.]

=

[Substitute S from step1.]

= 9

[Simplify.]

The velocity of the particle at

Correct answer : (3)

9.

The displacement S of a particle in time $t$ seconds is given by S = 7$t$^{3} - 3$t$^{2} - 4$t$ + 3 ft . What is the acceleration of the particle at time $t$?

a. | (42$t$ - 6) ft/s ^{2} | ||

b. | (42$t$ + 6) ft/s ^{2} | ||

c. | (42$t$) ft/s ^{2} | ||

d. | (- 42$t$ - 6) ft/s ^{2} |

The velocity of the particle at time

[Definition.]

=

[Substitute S from step1.]

Þ V = (21

The acceleration of the particle at time

[Definition.]

=

[Substitute V from step4.]

Þ

[Simplify.]

Correct answer : (1)

10.

What is the acceleration of a body at $t$ = 4 sec whose displacement S(in metres) is related with the time as S = 3$t$^{3} - 5$t$^{2} - 4$t$ + 2?

a. | - 62 m/sec ^{2} | ||

b. | - 72m/sec ^{2} | ||

c. | 72 m/sec ^{2} | ||

d. | 62 m/sec ^{2} |

The velocity of the particle at time

[Definition.]

=

[Substitute S from step1.]

Þ V = 9

The acceleration of the body at time

=

[Substitute V from step4.]

Þ

At

Correct answer : (4)