Basic Differentiation and Integration for Motion

Section A: Foundations

Question 1mcq

If

x=4t+3

, the velocity is?

A.

4\,m/s

B.

4t\,m/s

C.

3\,m/s

D.

7\,m/s

Question 2mcq

The slope of a position-time graph represents?

A.Acceleration

B.Velocity

C.Displacement

D.Mass

Question 3mcq

If

v=6t

, the acceleration is?

A.

6t\,m/s^2

B.

t/6\,m/s^2

C.

6\,m/s^2

D.

0

Question 4mcq

The derivative of a constant position is?

A.

1

B.The constant itself

C.Undefined

D.

0

Question 5mcq

If

x=t^2

, then

v

is?

A.

2t

B.

t

C.

t^3

D.

2

Question 6mcq

If

v=3t^2

, then

a

is?

A.

3t

B.

6t

C.

6t^2

D.

t^3

Question 7mcq

The area under a velocity-time graph gives?

A.Acceleration

B.Jerk

C.Displacement

D.Mass

Question 8mcq

If velocity is constant at

5\,m/s

for

4\,s

, displacement is?

A.

9\,m

B.

1.25\,m

C.

25\,m

D.

20\,m

Question 9mcq

Which expression is a product suitable for the product rule?

A.

x=t(t+2)

B.

x=t+2

C.

x=t^2

D.

x=5

Question 10mcq

Which expression is a composite function suitable for a basic chain rule?

A.

x=2t+1

B.

x=(2t+1)^3

C.

x=t^3+1

D.

x=7

Section B: Core Applications

Question 11mcq

For

x=3t^2+2t-5

, the velocity at

t=2\,s

is?

A.

10\,m/s

B.

18\,m/s

C.

14\,m/s

D.

9\,m/s

Question 12mcq

If

x=t^3-6t^2+9t

, at what time is the particle instantaneously at rest?

A.

t=0\,s

only

B.

t=2\,s

only

C.

t=1\,s

only

D.

t=1\,s

or

3\,s

Question 13mcq

For

v=4t^2-2t+1

, acceleration at

t=3\,s

is?

A.

22\,m/s^2

B.

10\,m/s^2

C.

24\,m/s^2

D.

35\,m/s^2

Question 14mcq

A particle has

x=t(t+4)

. Using the product rule,

v

equals?

A.

t+4

B.

2t+4

C.

t^2+4

D.

2t

Question 15mcq

If

x=(3t+1)^2

, then

v

is?

A.

2(3t+1)

B.

9t^2+1

C.

6(3t+1)

D.

6t+1

Question 16mcq

Velocity varies as

v=2t+3

. The displacement from

t=0

to

t=4\,s

is?

A.

20\,m

B.

32\,m

C.

44\,m

D.

28\,m

Question 17mcq

A velocity-time graph is a straight line from

v=2\,m/s

at

t=0

to

v=10\,m/s

at

t=4\,s

. Displacement is?

A.

24\,m

B.

16\,m

C.

20\,m

D.

40\,m

Question 18mcq

For

x=5-2t+t^2

, when does velocity become zero?

A.

0.5\,s

B.

1\,s

C.

2\,s

D.

5\,s

Question 19mcq

If

a=6t

and

v=4\,m/s

at

t=0

, then

v

at time

t

is?

A.

v=6t^2+4

B.

v=3t+4

C.

v=3t^2+4

D.

v=t^3+4

Question 20mcq

If

v=12-3t

, at what time does the particle stop?

A.

3\,s

B.

6\,s

C.

12\,s

D.

4\,s

Question 21mcq

A student differentiates

x=2t^3

and writes

v=6t^3

. What is the error?

A.The power should reduce by one, so

v=6t^2

.

B.The coefficient should remain

2

.

C.Velocity should be

t^2

.

D.The derivative is zero.

Question 22mcq

For

x=(t^2+1)(t+2)

, which expression correctly gives velocity using the product rule?

A.

v=2t(t^2+1)

B.

v=2t(t+2)+(t^2+1)

C.

v=(2t+1)(t+2)

D.

v=t^2+2t+1

Question 23mcq

For

x=(t^2+4)^2

, velocity is?

A.

2(t^2+4)

B.

4t^2+8

C.

4t(t^2+4)

D.

2t(t^2+4)

Question 24mcq

If

v=t^2+2

, displacement from

t=1

to

t=3

is?

A.

10\,m

B.

12\,m

C.

\frac{26}{3}\,m

D.

\frac{38}{3}\,m

Question 25mcq

A velocity-time graph lies below the time axis for an interval. What does its area represent?

A.Negative displacement

B.Negative distance

C.Zero acceleration only

D.Positive displacement

Section C: Advanced Reasoning

Question 26mcq

For

x=t^4-4t^3

, at what non-zero time is acceleration zero?

A.

t=1\,s

B.

t=2\,s

C.

t=3\,s

D.

t=4\,s

Question 27mcq

A particle has

x=t^3-3t^2-9t

. At

t=4\,s

, is it speeding up or slowing down?

A.Slowing down

B.At rest

C.Speeding up

D.Cannot be determined

Question 28mcq

For

x=t^2(t+1)

, velocity at

t=2\,s

is?

A.

12\,m/s

B.

18\,m/s

C.

20\,m/s

D.

16\,m/s

Question 29mcq

If

x=(2t^2+1)^3

, velocity is?

A.

12t(2t^2+1)^2

B.

6t(2t^2+1)^2

C.

3(2t^2+1)^2

D.

12t^2(2t^2+1)

Question 30mcq

A particle has

v=6t-2t^2

. What is its total displacement from

t=0

until it first stops again?

A.

9\,m

B.

18\,m

C.

27\,m

D.

36\,m

Question 31mcq

Velocity is

v=t-2

. What is the distance travelled from

t=0

to

t=5\,s

?

A.

\frac52\,m

B.

\frac92\,m

C.

\frac{13}{2}\,m

D.

\frac{15}{2}\,m

Question 32mcq

If

a=4-2t

and

v(0)=3\,m/s

, what is

v(3)

?

A.

3\,m/s

B.

9\,m/s

C.

12\,m/s

D.

6\,m/s

Section D: Challenge Set

Question 33mcq

For

x=(t+1)(t^2-2t+3)

, velocity at

t=1\,s

is?

A.

2\,m/s

B.

0

C.

4\,m/s

D.

6\,m/s

Question 34mcq

For

x=\sqrt{2t+1}

, velocity is?

A.

\sqrt{2t+1}

B.

\frac{1}{\sqrt{2t+1}}

C.

\frac{2}{\sqrt{2t+1}}

D.

\frac{1}{2\sqrt{2t+1}}

Question 35mcq

A particle has

v=3t^2-12t+9

. During which interval is it moving in the negative direction?

A.

t<1

only

B.

t>3

only

C.

1<t<3

D.

0<t<1

Question 36mcq

The acceleration is

a=6t

. If

v(0)=0

and

x(0)=2\,m

, what is

x(t)

?

A.

x=3t^2+2

B.

x=6t^2+2

C.

x=t^3

D.

x=t^3+2

Question 37mcq

A velocity graph is triangular: it rises from

0

to

12\,m/s

in

3\,s

and falls to

0

by

7\,s

. Displacement is?

A.

42\,m

B.

21\,m

C.

36\,m

D.

84\,m

Question 38mcq

A particle has

x=t^3-6t^2+9t

. At which time does it change direction first after

t=0

?

A.

2\,s

B.

1\,s

C.

3\,s

D.

6\,s

Question 39mcq

If

x=(t^2+1)(t^2-1)

, the most efficient derivative is?

A.

v=2t(t^2-1)

B.

v=4t^2

C.

v=4t^3

D.

v=2t^3

Question 40mcq

For

v=|t-2|

from

t=0

to

t=4

, displacement is?

A.

0

B.

2\,m

C.

8\,m

D.

4\,m