A particle moves in a straight line such that its position in metres from a fixed origin is given by s ( t ) = t 3 - 3 t + 1 , where t is the time in seconds, t ≥ 0 . Find the initial position, velocity and acceleration. Hence describe the particle's motion at this instant.

Given that, a particle moves in a straight line such that its position in metres from a fixed origin is given by s ( t ) = t 3 - 3 t + 1 , where t ≥ 0 [Seconds] We know that, Velocity v t is the derivative of the displacement function s t and Acceleration a t , is the derivative of the velocity v t . i.e., v t = d s d t or s ' t and a t = d v d t = d 2 s d t 2 or s ' ' t . Now, differentiate s t with respect to t , then we get: d s d t = v t = 3 t 2 - 3 Again differentiate v t with respect to t , then we get: d 2 s d t 2 = a t = 6 t Now, the initial position of displacement is s 0 = 1 m And also, the initial position of velocity is v 0 = - 3 m / s Again, the initial position of Acceleration is a 0 = 0 At this instant, the particle is 1 metre from the origin in the positive direction, travelling towards the origin at - 3 m / s , and is not accelerating.

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A $150 m$ length of wire is cut into two pieces. One of the pieces is bent to form a square with side length $s$ , and the other piece is bent to form a rectangle with a side length that is twice its width. Find the value of $s$ which minimizes the sum of the areas enclosed by the square and rectangle.

[Enter the value excluding units]

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Important Questions on Measuring Change: Differentiation

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5Q>Q 5

An architect wants to design a rectangular water tank with an open top that can hold a capacity of $10 m^{3}$ . The materials needed for the tank cost $$ 10$ per square metre for the base, and $$ 6$ per square metre for the sides. The length of the base must be twice its width. Find the minimum cost of the material required to build the container. Give your answer to the nearest dollar.

[Enter the value excluding units]

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 1

A particle moves in a straight line such that its position in metres from a fixed origin is given by

s (t) = t^{3} - 3 t + 1

, where

t

is the time in seconds,

t \geq 0

. Find expressions for the particle's velocity and acceleration at time

t

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Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 1

A particle moves in a straight line such that its position in metres from a fixed origin is given by

s (t) = t^{3} - 3 t + 1

, where

t

is the time in seconds,

t \geq 0

. Find the initial position, velocity and acceleration. Hence describe the particle's motion at this instant.

MEDIUM

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 1

A particle moves in a straight line such that its position in metres from a fixed origin is given by

s (t) = t^{3} - 3 t + 1

, where

t

is the time in seconds,

t \geq 0

. Describe the motion of the particle at

t = 2

seconds.

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Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 1

A particle moves in a straight line such that its position in metres from a fixed origin is given by

s (t) = t^{3} - 3 t + 1

, where

t

is the time in seconds,

t \geq 0

. At what time(s) does the particle reverse direction?

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Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 1

A particle moves in a straight line such that its position in metres from a fixed origin is given by

s (t) = t^{3} - 3 t + 1

, where

t

is the time in seconds,

t \geq 0

. For what time interval is the particle speeding up?

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IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 1

A particle moves in a straight line such that its position in metres from a fixed origin is given by

s (t) = t^{3} - 3 t + 1

, where

t

is the time in seconds,

t \geq 0

. What is the total distance travelled by the particle between

t = 0

and

t = 3

seconds?

EASY

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IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 5 - Measuring Change: Differentiation>Exercise 5R>Q 2

When a ball is thrown straight up in the air its height above the ground is given by

s (t) = 12 t - t^{3} + 1

metres, where

t

is in seconds,

t \geq 0

. From what height above the ground was the ball released?

Important Questions on Measuring Change: Differentiation

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