Variable Acceleration in 2D (Edexcel A Level Maths: Mechanics): Exam Questions

[In this question position vectors are given relative to a fixed origin ]

At time seconds, where , a particle, , moves so that its velocity  ms^-1 is given by

Find the acceleration of when .

When , the position vector of is m.

Find the position vector of when .

[In this question, position vectors are given relative to a fixed origin.]

At time seconds, where , a particle has velocity ms^–1 where

Find the speed of at time seconds.

Find an expression, in terms of , and , for the acceleration of at time seconds, where .

At time seconds, the position vector of is m.

Find the position vector of at time second.

[In this question position vectors are given relative to a fixed origin .]

At time seconds, where , a particle moves so that its velocity is given by

When , the particle is at the origin .

Find the position vector of at time seconds.

Find the distance of the particle from at time , giving your answer to three significant figures.

[In this question, position vectors are given relative to a fixed origin .]

The velocity of a particle at time seconds, where is given by

Find the initial acceleration of the particle.

The particle’s initial position is at the point .

Find the distance of the particle from the origin at time seconds, giving your answer to three significant figures.

[In this question, position vectors are given relative to a fixed origin .]

The position vector of a boat, sailing on a lake is

where time is measured in hours and .

Show that the boat takes hours until it first returns to .

Find the velocity of the boat at time  hours.

[In this question, position vectors are given relative to a fixed origin .]

A particle moves in a plane with velocity, , at time seconds where . is given by

Find the acceleration of the particle at time  seconds.

Given that its initial position is at the origin, find the position vector of the particle at time seconds.

[In this question and are horizontal unit vectors.]

An aircraft is modelled as a particle moving in a horizontal plane. At time seconds, where , the acceleration of , , is given by

When , the velocity of is .

Find the velocity of at time seconds.

Find the speed of at time  seconds, giving your answer in kilometres per hour to three significant figures.

A remote-controlled car is driven around a playground with velocity, , at time seconds, given by

Find an expression for the displacement of the remote-controlled car,  metres, measured from its initial position.

The remote-controlled car starts from the point metres.

Find the position vector of the particle from the origin at time seconds.

Find the distance of the remote-controlled car from the origin after 40 seconds.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A spider is modelled as a particle moving on a horizontal floor. At time seconds, where , the acceleration of , , is given by

The velocity of at time is .

Find the speed of at time .

Find the position vector of at time seconds.

[In this question, position vectors are given relative to a fixed origin .]

At time seconds, where , a particle moves so that its position vector metres is given by

Explain why never reaches the origin .

Find an expression for the velocity of at time seconds.

Find the magnitude of acceleration of at time seconds, giving your answer to three significant figures.

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 , find the position vector of the particle, , at time seconds.

Find the time at which the particle has zero acceleration in the horizontal direction.

[In this question, position vectors are given relative to a fixed origin .]

At time seconds, where , a particle moves so that its position vector metres is given by

Find the velocity of at time seconds.

Find the value of at the instant when is moving in a direction parallel to .

Show that never moves in a direction perpendicular to .

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

At time seconds, where , a particle is moving in a horizontal plane with velocity given by

Find the acceleration of at time seconds.

Explain why the acceleration of in the direction of is constant.

Find the value of , other than , for which the acceleration of in the direction of is zero.

When , is at the point with position vector .

Find the position vector of at time seconds.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

An ice skater is modelled as a particle moving across a straight section of a frozen river. At time seconds, where , the position vector of , metres, is given by

Find the speed of at time , giving your answer to 3 significant figures.

Show that the acceleration of is constant and find the magnitude of this acceleration, giving your answer to 3 significant figures.

At time seconds, where , a particle moves in the - plane in such a way that its velocity ms⁻¹ is given by

When , is at the point and when , is at the point .

Find the exact distance .

At time seconds, where , a particle moves so that its acceleration ms⁻² is given by

At the instant when , the velocity of is ms^-1.

Find the velocity of when .

Find the value of at the instant when is moving in a direction perpendicular to .

At time seconds, where , a particle moves so that its position vector metres, relative to a fixed origin , is given by

Find the value of at the instant when the speed of is 5 ms⁻¹.

At time seconds, where , a particle has velocity ms^–1 where

Find the speed of at time .

Find the value of when is moving parallel to .

Find the acceleration of at time seconds.

Find the value of when the direction of the acceleration of is perpendicular to .

At time seconds, a particle has velocity ms⁻¹, where

Find the acceleration of at time seconds, where .

Find the value of at the instant when is moving in the direction of .

At time seconds, where , the position vector of , relative to a fixed origin , is metres.

When , .

Find an expression for in terms of .

Find the exact distance of from at the instant when is moving with speed 10 ms⁻¹.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A boat is modelled as a particle moving on a lake. At time seconds, where , the position vector of , metres, is given by

Show that the distance of from when is .

Show that the speed of at the instant it first returns to is .

[In this question and are horizontal unit vectors.]

An aircraft is modelled as a particle moving in a horizontal plane. Once reaches its cruising height, at time hours where , the acceleration of , , is given by

The velocity of at time is .

Find the velocity of at time hours.

Find the speed of when .

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A remote-controlled car is modelled as a particle moving in a horizontal plane. At time seconds, where , the velocity of , , is given by

When , the position vector of is .

(i) Find the position vector of at time seconds.

(ii) Find the distance of from at the instant when .

At the same time as the car starts moving, a remote-controlled truck, modelled as a particle , is set in motion. The position vector of , ​ metres, at time seconds is given by

Determine the time at which the car and the truck collide.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A spider is modelled as a particle moving on a horizontal floor. At time seconds after emerging from under a skirting board, where , the acceleration of , , is given by

When , the velocity of is .

Find the velocity of at time seconds.

When , the position vector of is .

Find the distance of from at the instant it emerges from under the skirting board.

[In this question is a horizontal unit vector and is a vertical unit vector directed upwards. Position vectors are given relative to a fixed origin .]

A stone is modelled as a particle thrown from a point on the edge of a deep hole. At time seconds after being thrown, where , the velocity of , , is given by

Find the position vector of at time and give an interpretation of the component of this position vector.

The hole has a depth of 384 m.

Find the magnitude of the acceleration of at the instant it hits the bottom of the hole.

A particle’s velocity is modelled by the equation

where is the time in seconds.

The particle’s initial displacement is .

Find an expression for , at time seconds. State appropriate units with your answer.

Find when . State appropriate units with your answer.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

At time seconds, where , a particle moves so that its acceleration, , is given by

When , the velocity of is and is at the origin .

Find the distance of from when .

Find the value of at the instant when is moving in the direction of .

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin on one bank of a straight river.]

An ice skater is modelled as a particle moving across the river. The banks of the river are parallel to the vector . At time seconds, where , the position vector of , metres, is given by

Find the distance of from when .

Given that the skater reaches the opposite bank of the river when , find the width of the river.

Show that the magnitude of the acceleration of at time seconds is given by .

[In this question is a horizontal unit vector and is a vertical unit vector directed upwards. Position vectors are given relative to a fixed origin at the edge of the hole.]

A stone is modelled as a particle projected from . At time seconds after projection, where , the velocity of , , is given by

Find the position vector of at the instant it returns to the level of .

Find the maximum height of above .

The deepest known cave in the world, the Veryovkina Cave in Georgia, has a depth of 2212 m. The model predicts that the stone would take approximately 28 seconds to reach this depth.

Calculate the average vertical speed and the magnitude of the acceleration of at and use these values to comment on the validity of the model.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A particle moves in a horizontal plane. At time seconds, where , the velocity of , , is given by

When , the position vector of is .

Find at time seconds.

Find the value of at the instant when the acceleration of is parallel to .

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A boat is modelled as a particle moving on a lake. At time seconds, the displacement of , metres, relative to a fixed mooring point is given by

The position vector of relative to is .

(i) Write down an expression for the position vector of relative to at time seconds.

(ii) Find the difference between the distance of from and the distance of from at the instant when .

(i) Find the velocity of at time seconds.

(ii) Given that one trip around the lake takes 30 minutes, find the values of in the interval for which is moving parallel to one of the coordinate axes.

[In this question and are horizontal unit vectors.]

An aircraft is modelled as a particle moving in a horizontal plane. Once reaches its cruising height, at time hours where , the acceleration of , , is given by

When , the velocity of is .

Given that:

• the component of is positive and is double the component of ,

• and the speed of is ,

find the velocity of at time seconds.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

Two remote-controlled cars, coloured red and blue, are modelled as particles and moving on a horizontal playground. At time seconds, where , the velocity of , , and the velocity of , , are given by

When , the position vector of is and the position vector of is .

Find the value of , where , when and are equidistant from and find the distance of the cars from at this instant.

Hence show that and do not collide at this instant.

[In this question and are horizontal unit vectors and position vectors are given relative to a fixed origin .]

A spider is modelled as a particle moving on a horizontal floor. The floor is represented by the rectangle , shown below, defined by and , where the units are metres and the origin is a corner of the room.

At time seconds after emerges from , where , the acceleration of , , is given by

The velocity of at time seconds is .

Given that:

• the component of when is twice the component of when ,

• and the component of when is three times the component of when .

Find an expression for in terms of .

Determine whether is within the region when .

The position vectors in the x-y plane of two particles, A and B, at time t, are given by

(i) Write down the initial position of both particles.

(ii) Briefly explain what happens to the position of both particles for very high values of t.

(i) Find the velocity of particle A in terms of t.

(ii) Hence write down the velocity of particle B in terms of t.

(i) On the same diagram sketch the graphs of y against x for both r_A and  r_B.

(ii) Using your graph, and without doing any calculations explain why for all values of t,