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

The position vector of a boat, sailing on a lake relative to an origin, is

where time  is measured in hours.

(i) Show that the boat is initially at the origin.

(ii) Show that the boat takes  hours until it first returns to the origin.

(i) Find the velocity,  km h⁻¹, of the boat, at time  hours.

(ii) Find the velocity of the boat at time   hours.

A particle moving in a plane has velocity, , at time  seconds, given by 

(i) Find an expression for the acceleration of the particle.

(ii) Find the acceleration of the particle at time   seconds.

(i) Find the position vector of the particle given that its initial position is at the origin.

(ii) Find the position vector of the particle at time seconds.

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.

An ice skater moves across an ice rink such that their position, at time seconds relative to an origin, is given by 

(i) Briefly explain how you can tell the ice skater’s motion did not start at the origin.

(ii) Find the coordinates of the position of the ice skater after 40 seconds.

(iii) Find the distance between the ice skater and the origin after 40 seconds.

(i) Find an expression for the velocity of the ice skater at time  seconds.

(ii) Find an expression for the acceleration of the ice skater at time  seconds.

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.

(i) Find the distance of the remote-controlled car from its initial position after 40 seconds.

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

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. 

(i) Given that the spider’s initial velocity was  , find the velocity of the spider at time  seconds.

(ii) Find the speed of the spider after 3 seconds.

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.

The position vector of a particle at time  seconds is given by

(i) Write down the initial position of the particle.

(ii) Briefly explain why the particle gets closer to the origin but never actually meets it.

Find an expression for the velocity of the particle at time seconds.

(i) Find an expression for the acceleration of the particle at time seconds.

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

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.

(i) Find the acceleration of the stone at time seconds.

(ii) Find the magnitude of the acceleration of the stone at time  seconds.

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.

(i) Find the acceleration of the particle, at time  seconds.

(ii) Find the time at which the particle has no acceleration in the horizontal direction.

At time seconds, a particle has position vector  m, where 

Find the velocity of at time seconds, where .

(i) Find the value of at the instant when  moves in a direction parallel to .

(ii) Show that  will never move in a direction perpendicular to .

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.

(i) Show that the boat is initially at the mooring point.

(ii) Show that the distance from the mooring point to the boat at time  is .

(i) Find the velocity,  m s⁻¹, of the boat, at time  seconds.

(ii) Show that the boat has a speed of  when it first returns to the mooring point.

A particle moving in the   plane has velocity,  m s⁻¹, at time  seconds, given by

(i) Find the acceleration,  m s⁻², of the particle at time and explain how you can tell that the acceleration in the -direction is constant.

(ii) Other than, find the time at which the acceleration in the direction is zero.

Find the position vector of the particle given that its initial position is at the point 

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.

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.

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

(i) The remote-controlled car is initially set in motion from position . Find the position vector of the car at time  seconds.

(ii) Find the distance of the remote-controlled car from the origin after 15 seconds.

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.

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.

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.

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 .

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.

(i) Write down the position vector,  m, of the boat at time seconds.

(ii) Find the difference between the distance the boat is from its mooring point and the distance it is from the origin at the point when  seconds.

(i) Find the velocity,  m s⁻¹, of the boat, at time seconds.

(ii) Given that one trip around the lake takes half an hour, find the times at which the boat is moving in one direction only during one trip around the lake.

Once an aircraft reaches its cruising height (at time  hours) its acceleration is modelled by

Given that at time  hours,

• the i-component of the aircraft’s velocity is positive and double that of the j‑component, and,

• the speed of the aircraft is ,

 find the velocity of the aircraft in terms of .

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 .

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

• its speed in the i-direction after 3 seconds is twice the speed in the i-direction after 2 seconds,

• its speed in the j-direction after 3 seconds is three-times the speed in the j-direction after 1 second,

• neither component of the spider’s velocity is ever negative.

 Determine whether or not the spider is still in the room (and visible, so not under the skirting board) after 6 seconds.

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.

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.

The position vectors in the  plane of two particles, and , at time ,  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 .

(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 ,