A particle’s position, at time seconds, is given by the vector
Consider the -coordinate of the particle to explain why the particle will never pass through the origin.
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A particle’s position, at time seconds, is given by the vector
Consider the -coordinate of the particle to explain why the particle will never pass through the origin.
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At time seconds, a particle moving in a plane has velocity
Use differentiation to find an expression for the acceleration of the particle.
Use integration to find the displacement of the particle from its initial position.
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The acceleration of a particle is modelled using
where time is measured in seconds.
Given that the particle is initially at rest, use integration to find an expression for the velocity of the particle.
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The position vector of a particle, at time seconds, is given by
Differentiate with respect to to find an expression for the velocity of the particle.
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The velocity of a particle at time seconds is given by
Given that the particle’s motion began at the origin, use integration to find the position vector of the particle at time seconds.
Use Pythagoras’ theorem to find the distance of the particle from the origin at time second, giving your answer to three significant figures.
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The acceleration of a particle is modelled using the equation
where time is measured in seconds.
(i) Use integration to find an expression for the velocity of the particle.
(ii) Given that when , find the value of the constant(s) of integration.
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The motion of a particle, starting from the origin, is described by the position vector
where time is measured in seconds.
Differentiate with respect to twice to find an expression for the acceleration of the particle.
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The velocity of a particle at time seconds is given by
Differentiate to find the acceleration, , of the particle at time seconds.
Use Pythagoras’ theorem to find the magnitude of acceleration of the particle at time seconds, giving your answer to three significant figures.
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A particle, starting from rest at the origin, has acceleration
at time seconds.
Integrate with respect to twice to find an expression for the position vector of the particle. Remember to account for any constant(s) of integration.
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The velocity of a particle at time seconds is given by
The particle’s initial position is at the point ( 4 , 5 ) .
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The velocity of a particle at time seconds is given by
Differentiate to find the acceleration, m s-2, of the particle at time seconds.
Integrate to find the position vector, m, of the particle at time seconds given that its initial position is the origin.
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The position vector of a boat, sailing on a lake relative to an origin, is
where time is measured in hours.
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A particle moving in a plane has velocity, , at time seconds, given by
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Once an aircraft reaches its cruising height (at time seconds) its acceleration is modelled by
Given that the velocity of the aircraft at is , find the velocity of the aircraft in terms of .
Find the speed of the aircraft at time seconds, giving your answer in kilometres per hour to three significant figures.
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An ice skater moves across an ice rink such that their position, at time seconds relative to an origin, is given by
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A remote-controlled car is driven around a playground with velocity, m s−1, at time seconds, given by
Find an expression for the displacement of the remote-controlled car, m, from its initial position.
The remote-controlled car is set in motion from the point . Find the position vector of the particle at time seconds.
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A spider is crawling across the floor of a house such that is has acceleration
at time seconds after the spider emerged from under the skirting board.
The point at which the spider emerged from under the skirting board is deemed the origin. Find the position vector of the spider at time seconds.
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The position vector of a particle at time seconds is given by
Find an expression for the velocity of the particle at time seconds.
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A stone is thrown from the edge of a deep cave such that it will fall into the cave with its motion described by the equation
where is the velocity of the particle seconds after the stone is thrown.
Find the speed at which the stone is thrown.
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A particle’s velocity is modelled by the equation
where is the time in seconds.
Given that the particle is initially located at the point (2 , 1), find the position vector of the particle, m, at time seconds.
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At time seconds, a particle has position vector m, where
Find the velocity of at time seconds, where .
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The position of a boat on a small lake, relative to a mooring point located at the origin, is given by the vector
where time is measured in seconds.
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A particle moving in the plane has velocity, m s−1, at time seconds, given by
Find the position vector of the particle given that its initial position is at the point
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Once an aircraft reaches its cruising height (at time hours) its acceleration is modelled by
Given that the velocity of the aircraft at hours is , find the velocity of the aircraft in terms of .
Find the speed of the aircraft when it first reaches its cruising height.
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An ice skater moves across a straight section of a frozen river such that their position, at time seconds relative to an origin is given by
Find the initial speed of the ice skater giving your answer to three significant figures.
Show that the ice skater’s acceleration is constant and find the magnitude of the acceleration, giving your answer to three significant figures.
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A remote-controlled car is driven around a large playground with velocity, , at time seconds, given by
At the same time as the remote-controlled car is started, a remote-controlled truck is also set into motion. The truck has position vector, m, at time seconds given by
Determine the time(s) at which the car and the truck will collide, if at all.
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A spider is crawling across the floor of a house such that is has acceleration
at time seconds after the spider emerged from under the skirting board.
After 3 seconds the spider’s velocity is .
Find the velocity of the spider at time seconds.
After 3 seconds the spider’s position, relative to an origin at a corner of the floor, is . Find the distance the spider is from the origin when it emerges from under the skirting board.
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A particle’s velocity is modelled by the equation
where is the time in seconds.
The particle’s initial displacement is .
Find the position vector of the particle, m, at time seconds.
Find the magnitude of the acceleration of the particle after 1 second.
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At time seconds, a particle has acceleration m s−2, where
.
Initially starts at the origin and moves with velocity .
Find the distance between the origin and the position of when .
Find the value of at the instant when is moving in the direction of .
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The displacement of a boat, m on a small lake, relative to a mooring point located at the point (10, 0), is given by the vector
where time is measured in seconds.
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Once an aircraft reaches its cruising height (at time hours) its acceleration is modelled by
Given that at time hours,
find the velocity of the aircraft in terms of .
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An ice skater moves across a straight section of a frozen river such that their position, at time seconds relative to an origin is given by
(i) Find the distance the skater is from the origin after 25 seconds.
(ii) As they skate forwards, the skater slowly crosses the width of the river.
It takes 225 seconds for the skater to cross the river.
How wide is the river?
Show that the magnitude of acceleration of the skater at time seconds is given by .
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A spider is crawling across the floor of a house such that its acceleration is
at time t seconds after the spider emerged from under the skirting board in a corner of the room.
The room measures 12 m × 8 m and the corner from which the spider emerges is the origin, as shown below.
The spider’s velocity is such that
Determine whether or not the spider is still in the room (and visible, so not under the skirting board) after 6 seconds.
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A stone is thrown from the edge of a deep cave such that it will fall into the cave and has velocity
at time seconds after it is thrown.
Find the position of the stone at the time it is about to fall into the cave (rather than being in the air above the cave).
Find the maximum height above the cave the stone reaches.
The deepest known cave in the world has a depth of 2212 m. (The Veryovkina Cave in Georgia.) The model above suggests the stone would take around 28 seconds to reach this depth. Consider the average speed and the acceleration of the stone to highlight a problem with this model.
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A particle’s velocity is modelled by the equation
where is the time in seconds.
The particle’s initial position is (3 , 5), find the position vector of the particle, m, at time seconds.
Find the time at which the particle’s acceleration, is zero in the horizontal direction.
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The position vectors in the plane of two particles, and , at time , are given by
(i) Find the velocity of particle in terms of .
(ii) Hence write down the velocity of particle in terms of .
(i) On the same diagram sketch the graphs of against for both and .
(ii) Without doing any calculations explain why for all values of ,
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