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In this lesson, we will learn how to use differentiation to get the average and instantaneous velocities and acceleration vectors of a particle in straight-line motion.

Q1:

A particle is moving in a straight line. After time π‘ seconds, where π‘ β₯ 0 , the bodyβs displacement relative to a fixed point is given by β π = ο οΌ 5 6 π‘ + 5 π‘ ο β π ο 3 m , where β π is a fixed unit vector. Find the initial velocity of the particle β π£ 0 and its acceleration β π , 5 seconds after it started moving.

Q2:

A particle moves along the π₯ -axis such that at time π‘ seconds its displacement from the origin is given by Determine the particleβs acceleration when π₯ = β 5 m .

Q3:

A stone is projected vertically upwards. At time π‘ seconds, its height from the ground is given by Determine the speed of the stone when it is 22.5 m high.

Q4:

A particle moves along the π₯ -axis. When its displacement from the origin is π m, its velocity is given by Find the particleβs minimum velocity.

Q5:

A particle moves along the π₯ -axis such that at time π‘ seconds its velocity is given by After how many seconds is its acceleration equal to 0?

Q6:

A particle moves along a straight line. Its displacement at time π‘ is π₯ = β ( π‘ ) c o s . Which of the following statements about the acceleration of the particle is true?

Q7:

A particle started moving along the π₯ -axis. At time π‘ seconds, its position relative to the origin is given by Find the maximum distance between the particle and the origin π₯ m a x , and determine the velocity of the particle π£ when π‘ = 3 π s .

Q8:

A particle is moving in a straight line such that its speed π£ , measured in metres per second, and its position π₯ , measured in metres, satisfy the following equation: Find the maximum speed of the particle π£ m a x and the acceleration of the particle π when π£ = π£ m a x .

Q9:

A particle is moving in a straight line such that its displacement π after π‘ seconds is given by Find the velocity of the particle when its acceleration is zero.

Q10:

A particle is moving in a straight line. The relation between its velocity π£ , measured in metres per second, and its position π₯ , measured in metres, is given by Find the magnitude of its acceleration when its velocity is zero.

Q11:

A particle is moving in a straight line such that its velocity π£ at time π‘ seconds is given by Find the magnitude of the acceleration of the particle when its velocity is 94 m/s.

Q12:

A particle moves along a straight line. Its displacement at time π‘ is π₯ = ( π‘ ) c o s . Which of the following statements about the acceleration of the particle is true?

Q13:

A particle moves along a straight line. Its displacement at time π‘ is π₯ = π‘ s i n . Which of the following statements about the acceleration of the particle is true?

Q14:

A particle is moving in a straight line such that its displacement π₯ after π‘ seconds is given by Determine the time after which the particle changes its direction.

Q15:

A particle is moving in a straight line such that its displacement π₯ at time π‘ seconds is given by What distance does the particle travel in the first 9.6 seconds?

Q16:

A particle moves along the π₯ -axis. At time π‘ seconds, its displacement from the origin is given by Determine all the possible values of π‘ , in seconds, at which the particleβs speed β π£ β = 4 m/s.

Q17:

A particle is moving in a straight line such that its displacement from the origin after π‘ seconds is given by Find its velocity π£ when π‘ = π 4 s and its acceleration π when π‘ = π 3 s .

Q18:

A particle moves along the π₯ -axis. At time π‘ seconds, its displacement from the origin is given by When π‘ = 1 s , π₯ = 7 m , and when π‘ = 2 s , the particleβs velocity is 7 m/s. Determine the value of π β π .

Q19:

A particle moves along a straight line. Its displacement at time π‘ is π₯ = β π‘ t a n . Find its velocity, π£ , and hence determine which of the following expressions is equal to the acceleration of the particle.

Q20:

A particle moving along a path has velocity π£ and acceleration π . Given that the equation of the displacement is π₯ = π‘ t a n , find π .

Q21:

A particle moves along a straight line. Its displacement at time π‘ is π₯ = β ( π‘ ) s i n . Which of the following statements about the acceleration of the particle is true?

Q22:

A particle moves along the π₯ -axis such that at time π‘ seconds its displacement from the origin is given by Find the particleβs velocity, π£ , and acceleration, π , at π‘ = π s .

Q23:

A particle is moving in a straight line such that its velocity π£ and position π₯ satisfy the following equation: Find an expression for the particleβs acceleration in terms of π₯ .

Q24:

A particle started moving along the π₯ -axis. When the particleβs displacement relative to the origin was β π m in the direction of increasing π₯ , its velocity was β π£ = 5 β π / m s . Determine the particleβs acceleration when β π = 1 3 m .

Q25:

A particle moves in a straight line such that at time π‘ seconds, its velocity is given by After how many seconds does the direction of the particleβs motion change?

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