9.05 Differentiation and kinematics

The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 6 t^{2}

The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 3 t^{3} - 4 t^{2}

The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 3 t^{2} + 5 t + 2

The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 18 \sqrt{t}

The position (in metres) of a drag racer from the starting line at a given time t seconds can be modelled by the equation:

x \left( t \right) = \dfrac{t^{3}}{2} + 3 t^{2}

Sophia is throwing a ball straight up in the air. The height, h, above the point where the ball is released is given by the equation h = - 4.9 x^{2} + 11.27 x.

The height in metres of a projectile above level, flat ground is given by h = 9 + 8 t - t^{2}, where t is given in seconds.

A boy stands on the edge of a sea-cliff with a height of 48 \text{ m}. He throws a stone off the cliff so that its vertical height above the cliff is given by h = 16 t - 4 t^{2} where t is given in seconds.

The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by:

The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by: x \left( t \right) = \left( 4 t + 3\right)^{2}

The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by:

The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by: x \left( t \right) = - 2 t \left(t + 10\right)

Let s = 10 + 12 t - 4.7 t^{2} be the height of an object in metres at time t in seconds.

A boy throws a ball vertically. The height of the ball h in metres is given by h = 1 + 17 t - 5 t^{2} where t is given in seconds.

The displacement (in metres) of a body moving along a straight line after t seconds is modelled by: x \left( t \right) = - t^{3} + a t^{2} + b t + 7 The initial velocity of the body is 8 \text{ m/s}. The body is momentarily at rest at t = 2 seconds.

Find the values of a and b.

The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by:

The displacement (in metres) of a body from an origin O at time t seconds is modelled by: x \left( t \right) = t^{2} - 7 t + 5

A car starts at rest and has a displacement of s \text{ m} in t seconds, where s = \dfrac{1}{6} t^{3} + \dfrac{1}{4} t^{2}.

A car starts at rest and has a displacement of s \text{ m} in t seconds, where s = t^{4} + t^{2}.

A particle moves in a straight line and its displacement after t seconds is given by: x = 19 t^{2} - 2 t^{3} where x is its displacement in metres.

Let v and a represent its velocity and acceleration at time t respectively.

A particle travels in a straight line such that its displacement x in metres from a fixed point is given by: x = 12 + 6 t^{2} - t^{3} where t is the time in seconds after it starts moving.

A particle moves in a straight line, starting from rest at the point O. At time t seconds after leaving O, the speed v of the particle (in \text{m/s}) is given by: v \left( t \right) = \left( 3 t - 2 t^{2}\right)^{2}

Describe the motion of the particle during the first 2 seconds.

A particle moves in a straight line. Its velocity (in metres per second), t seconds after passing the origin is given by: v = 2 t^{2} - 10 t Find the velocity when the acceleration of the particle is zero.

The position of an object moving along the x-axis at time t seconds is given by: x = t^{3} - 6 t^{2} + 3 t - 2 Its velocity and acceleration at time t seconds are given by v and a respectively. Find the position x of the object when its acceleration is 0.

The velocity (in metres per second) of a body moving in rectilinear motion after t seconds is modelled by: v \left( t \right) = t^{2} - 11 t + 24

The velocity (in metres per second) of a body moving in rectilinear motion after t seconds is modelled by: v \left( t \right) = 5 t^{2} - 23 t + 24

An object, P, moves in rectilinear motion such that its displacement x (in metres) from the origin at time t seconds is given by: x \left( t \right) = \left(t + 1\right) \left(4 - t\right) \left(t + 4\right)Its velocity and acceleration at time t are given by x' \left( t \right) and x'' \left( t \right) respectively.

The distance d in metres travelled by a train between two stations is given by: d = 0.86 t^{2} - 0.0084 t^{3} where t is given in seconds.

The following table represents data for a typical driver approaching a red traffic light. At a given speed, the driver must first react and then press the brake. The distance travelled during this period is called the thinking distance. After the brakes are applied, the driver slows down to a stop and covers a distance called the braking distance.

The position of a particle relative to a fixed position O along a straight line is shown by the following graph, along with a tangent line at t = 8. If the particle is positioned to the right of the point O, then it has positive displacement.

The position of a particle relative to a fixed position O along a straight line is shown by the following graph. If the particle is positioned to the right of the point O, then it has positive displacement.

The position of a particle relative to a fixed position O along a straight line is shown by the following graph, along with a tangent line at t = 10. If the particle is positioned to the right of the point O, then it has positive displacement.

An object moves in a straight line, with its position from a point O given by the equation: x \left( t \right) = t^{3} - 4 t^{2} + 5

Find the total distance that the object has travelled by the time t = \dfrac{8}{3} units.

A particle, P, starts from rest at point O, and its velocity t seconds after it starts moving is modelled by the equation: v = t^{2} \left(9 - t\right) After 9 seconds it comes to rest and stops moving.

A particle moves in a straight line and its displacement after t seconds is given by: x = 12 t - 2 t^{2} where x is its displacement in metres from the starting point. Let v and a represent its velocity and acceleration at time t respectively.

Find the total distance that the particle has travelled in the first 9 seconds.

The position of a particle in metres relative to a point O along a straight line is given by: x \left( t \right) = t^{3} - 9 t^{2} + 24 t where t is in seconds.

A particle passes through a hole in a wall. It's displacement from the wall, s, metres at time t seconds after it passes through the hole is given by:s = \dfrac{1}{3} t + \dfrac{1}{9} t^{2} - \dfrac{1}{27} t^{3}In this case, t may be negative.