Projectiles (Edexcel A Level Maths: Mechanics): Exam Questions

A small ball is projected with speed $U$ ms⁻¹ from a point $O$ at the top of a vertical cliff.

The point $O$ is 25 m vertically above the point $N$ which is on horizontal ground.

The ball is projected at an angle of 45° above the horizontal.

The ball hits the ground at a point $A$, where $AN$ = 100 m, as shown in Figure 2.

The motion of the ball is modelled as that of a particle moving freely under gravity.

Using this initial model, show that $U=28$.

Find the greatest height of the ball above the horizontal ground $NA$.

In a refinement to the model of the motion of the ball from $O$ to $A$, the effect of air resistance is included.

This refined model is used to find a new value of $U$.

How would this new value of $U$ compare with 28, the value given in part (a)?

State one further refinement to the model that would make the model more realistic.

A small stone is projected with speed 65 ms⁻¹ from a point $O$ at the top of a vertical cliff.

Point $O$ is 70 m vertically above the point $N$.

Point $N$ is on horizontal ground.

The stone is projected at an angle $\alpha$ above the horizontal, where $\tan \alpha =\frac{5}{12}$.

The stone hits the ground at the point $A$, as shown in Figure 3.

The stone is modelled as a particle moving freely under gravity.

The acceleration due to gravity is modelled as having magnitude 10 ms⁻².

Using the model, find the time taken for the stone to travel from $O$ to $A$.

Using the model, find the speed of the stone at the instant just before it hits the ground at $A$.

One limitation of the model is that it ignores air resistance.

State one other limitation of the model that could affect the reliability of your answers.

A particle is projected from a platform 5 m vertically above ground level with velocity $\left(6\mathbf{i}+8\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

(i) Find the speed with which the particle is projected.

[1]

(ii) Find the angle above the horizontal at which the particle is projected.

[1]

Find the greatest height above the ground reached by the particle.

Find the time of flight of the particle.  Give your answer to three significant figures.

A particle is projected horizontally from a height of 15 m vertically above the ground with a speed of $8\mathrm{m}{\mathrm{s}}^{-1}$.

Find the time of flight of the particle.

Find the horizontal displacement of the particle when it reaches the ground.

A cannon points horizontally from the top of a castle wall.

A cannonball leaves the cannon with speed 150 ms^-1 and hits the ground 2.0 seconds later.

Find the horizontal range of the cannonball.

Find the height of the castle wall above the ground.

A golfer strikes a ball from ground level with velocity  $\left(35\sqrt{3}\mathbf{i}+35\mathbf{j}\right){\mathrm{ms}}^{-1}$.

Find

(i) the initial speed of the golf ball,

(ii) the angle from the horizontal at which the golf ball is struck.

Find the time taken from when the golf ball is struck until it reaches the ground for the first time. Assume the ground is horizontal.

Find the maximum height reached by the golf ball above the ground.

A stuntperson rides a bicycle off a straight ramp inclined 30° to the horizontal at the end of a pier.  

The bicycle leaves the ramp with a speed of $14\mathrm{m}{\mathrm{s}}^{-1}$.

The bicycle and stuntperson are modelled as a single particle and land in the water at a horizontal distance of $28\mathrm{m}$ from the end of the pier.

Calculate the time (in seconds) for which the stuntperson and bicycle are in the air above the water.

Given that the end of the ramp was 50 cm above the pier, find the height of the pier, giving your answer to the nearest tenth of a metre.

A particle is launched from the the ground at an angle of $80°$ above the horizontal with an initial speed of $75\mathrm{m}{\mathrm{s}}^{-1}$.

Find

(i) the initial horizontal speed of the particle,

(ii) the initial vertical speed of the particle.

Find the times at which the particle is at a height of 200 m and hence find the length of time for which the particle is above 200 m.

A hot-air balloon is hovering 20 m above a school sports field.

A student standing in the basket gently throws a small bean-bag to their friends on the ground.

The bean-bag leaves the balloon with speed 5 ms⁻¹, directed at 10° below the horizontal.

Assume that $g=9.8{\mathrm{ms}}^{-2}$ and that air resistance is negligible.

Calculate the time taken for the bean-bag to reach the ground.

Show that the bean-bag lands within 10 m of the point on the ground directly below the hot-air balloon.

An ejector seat for a small aircraft is being tested. The ejector seat is launched from a stationary position 2 m vertically above the ground.  

The seat is fired with an initial velocity of $25\mathrm{m}{\mathrm{s}}^{-1}$ at an angle $\alpha$ above the horizontal, where $\tan \alpha =\frac{24}{7}$.

To pass its first safety test the ejector seat must rise at least 16 m vertically above the position from which it was launched within 1 second.

Determine whether the seat passes its first safety test.

In another test, the ejector seat must deploy a parachute when it reaches its maximum height.

Find

(i) the height above the ground,

(ii) and the time after launch

at which the ejector seat should deploy its parachute.

A particle is projected from a point on a horizontal plane with initial velocity $U\mathrm{m}{\mathrm{s}}^{-1}$ at an angle of $\alpha °$ above the horizontal.  

The particle moves freely under gravity where $g\mathrm{m}{\mathrm{s}}^{-2}$ is the acceleration due to gravity.

Show that the time of flight of the particle, $T$ seconds, is given by

 $T=\frac{2U\sin \alpha }{g}$

Show that the range of the particle, $R$ m, on the horizontal plane is given by

$R=\frac{U^2\sin \left(2\alpha \right)}{g}$ 

In Toonland, a coyote is desperately trying to catch the very fast roadrunner bird.  In its latest effort to catch the roadrunner the coyote projects itself from a catapult at the top of a canyon which is 85 m high.  

The catapult projects the coyote with initial velocity $\left(3\mathbf{i}+9\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

The roadrunner spots the coyote’s plan when the coyote is at its maximum height above the ground.  Using magic Toon-paint the roadrunner paints a hole on the ground at the spot where the coyote will land.

It takes the roadrunner 4 seconds to paint the hole on the ground, and once it is finished the paint will become a real hole.  

Determine whether or not the roadrunner will succeed in causing the coyote to land in the hole.

A golf ball is at rest at the point $A$ on horizontal ground.

The ball is hit and initially moves at an angle $\alpha$ to the ground.

The ball first hits the ground at the point $B$, where $AB=120$ m, as shown in Figure 3.

The motion of the ball is modelled as that of a particle, moving freely under gravity, whose initial speed is $U$ ms^-1.

Using this model, show that $U^2\sin \alpha \cos \alpha =588$.

The ball reaches a maximum height of 10 m above the ground.

Show that $U^2=1960$.

In a refinement to the model, the effect of air resistance is included.

The motion of the ball, from $A$ to $B$, is now modelled as that of a particle whose initial speed is $V$ ms^–1.

This refined model is used to calculate a value for $V$.

State which is greater, $U$ or $V$, giving a reason for your answer.

State one further refinement to the model that would make the model more realistic.

A small ball is projected with speed 28 ms^–1 from a point $O$ on horizontal ground.

After moving for $T$ seconds, the ball passes through the point $A$.

The point $A$ is 40 m horizontally and 20 m vertically from the point $O$, as shown in Figure 2.

The motion of the ball from $O$ to $A$ is modelled as that of a particle moving freely under gravity.

Given that the ball is projected at an angle $\alpha$ to the ground, use the model to show that $T=\frac{10}{7\cos \alpha }$.

Show that ${\tan }^2\alpha -4\tan \alpha +3=0$.

Find the greatest possible height, in metres, of the ball above the ground as the ball moves from $O$ to $A$.

The model does not include air resistance.

State one other limitation of the model.

In Toon City, a coyote is desperately trying to catch the very fast roadrunner bird. 

In an effort to catch the roadrunner the coyote projects itself from a catapult at ground level.  The catapult projects the coyote with initial velocity $\left(15\mathbf{i}+8\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Modelling the coyote as a projectile find

(i) the initial speed of the coyote

(ii) the exact values of $\sin \alpha \mathrm{and}\cos \alpha$, where $\alpha$ is the angle above the horizontal at which the coyote is projected.

The variable $x$ can be used to represent the horizontal displacement whilst $y$ can be used to represent the vertical displacement.

Find the equation for the trajectory of the coyote in the form

$y=ax-\frac{g}{b}{x}^2$

where $a$ and $b$ are constants and $g$ is the acceleration due to gravity.

Find the horizontal range of the coyote's motion.

There is a cactus plant of height 4 m located exactly halfway along the trajectory of the coyote. 

Determine whether the coyote will collide with the cactus.

A particle is projected from a point $O$ on horizontal ground with speed $U$ ms⁻¹ at an angle of elevation $\beta$, where $\tan \beta =\frac{4}{3}$​. The particle moves freely under gravity and strikes the ground at a point $A$, where $OA$ = 120 m.

Show that $U=35$.

The particle passes through a point $P$ which is at a horizontal distance of 42 m from $O$.

Find the height of $P$ above the ground.

Find the direction of motion of the particle at the instant it passes through $P$, giving your answer as an angle in degrees relative to the horizontal.

A particle is projected horizontally from the point with coordinates $\left(0,5\right)$ with an initial speed of $9\mathrm{m}{\mathrm{s}}^{-1}$.  The coordinates are expressed in metres.

Throughout this question leave any coefficients in expressions and equations as exact values, given in terms of g where appropriate.

(i) Find, in terms of time t seconds, expressions for $s_x\mathrm{and}{s}_y$, the horizontal and vertical displacements of the particle from the point from which it was projected.

(ii) Write down an expression for $h_y$, the vertical displacement of the particle from the origin.

Find an equation for the trajectory of the particle in the form $y=\mathrm{f}\left(x\right)$.

A stuntperson aims to perform a motorcycle jump over a row of buses.  

The take-off and landing ramps are both at the same height, and the take-off ramp is angled at 20° above the horizontal.  

Each bus is 2.55 m wide, and the heights of the buses are less than the heights of the take-off and landing ramps.  

The stuntperson and motorcycle are modelled as a single particle.

If the stuntperson leaves the end of the launch ramp with a speed of $18\mathrm{m}\mathrm{s}{}^{-1}$, work out the maximum number of buses the stuntperson can clear, assuming they land on the landing ramp.

If the stuntperson wishes to jump over 16 buses using the same ramp, find the speed with which they should leave the ramp, giving your answer to three significant figures.

Deefa the dog is undergoing agility training, part of which involves jumping over a wall.  

The top of the wall sits 91 cm above the ground.  Deefa is modelled as a projectile jumping in a two-dimensional vertical plane that is perpendicular to the surface of the wall.

Deefa jumps with velocity $\left(3\mathbf{i}+4\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$, leaving the ground at a distance 75 cm horizontally from the wall.  

Determine whether Deefa will clear the wall with this jump.

On another attempt Deefa clears the wall by jumping with velocity $\left(4\mathbf{i}+5\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Deefa jumps at the latest possible moment in order to clear the wall.  

Find the distance between the wall and the point at which Deefa jumps off the ground.

The flight of a particle projected with an initial velocity of $U\mathrm{m}{\mathrm{s}}^{-1}$ at an angle $\alpha$ above the horizontal is modelled as a projectile moving under gravity only.  

The particle is projected from the point $\left(0,h\right)$ with the upward direction being taken as positive, and with the coordinates being expressed in metres. 

$g\mathrm{m}{\mathrm{s}}^{-2}$ is the constant of acceleration due to gravity.

Find, in terms of $U,\alpha ,h,g$ and time $t$ as appropriate, expressions for 

(i) the $x$-coordinate of the projectile at time $t$ seconds,

(ii) the $y$-coordinate of the projectile at time $t$ seconds.

For a particular projectile,

• $\sin \alpha =\frac{8}{17}$

• $U=51{\mathrm{ms}}^{-1}$

• and the particle is projected from the point $\left(0,6\right)$.  

Find an expression for the trajectory of the particle, giving your answer in the form $y=ax+bg{x}^2+c$ where $a,b\mathrm{and}c$ are rational constants.

For a particle modelled as a projectile with initial velocity $U\mathrm{m}{\mathrm{s}}^{-1}$ at an angle of $\alpha °$ above the horizontal, show that the equation of the trajectory of the particle is given by

$y=\left(\tan \alpha \right)x-\frac{g{x}^2}{2U^2{\cos }^2\alpha }$

In this question, use $\mathit{g}\mathbf{=}\mathbf{10}{\mathbf{ms}}^{\mathbf{-}\mathbf{2}}$ for the acceleration due to gravity

The graph below shows the trajectory of a projectile, with $x$ and $y$ being measured in metres.

Use the graph to help determine

(i) the time of flight of the projectile in seconds

(ii) the initial velocity of the projectile in the form $(u_x\mathbf{i}+u_y\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$

(iii) the speed, to three significant figures, of the projectile at launch

(iv) the angle to the horizontal at which the projectile was launched, giving your answer to one decimal place

(v) the maximum height reached by the projectile.

A boy throws a ball at a target. At the instant when the ball leaves the boy’s hand at the point $A$, the ball is 2 m above horizontal ground and is moving with speed $U$ at an angle $\alpha$ above the horizontal.

In the subsequent motion, the highest point reached by the ball is 3 m above the ground.

The target is modelled as being the point $T$, as shown in Figure 4.

The ball is modelled as a particle moving freely under gravity.

Using the model, show that $U^2=\frac{2g}{{\sin }^2\alpha }$.

The point $T$ is at a horizontal distance of 20 m from $A$ and is at a height of 0.75 m above the ground. The ball reaches $T$ without hitting the ground.

Find the size of the angle $\alpha$.

State one limitation of the model that could affect your answer to part (b).

Find the time taken for the ball to travel from $A$ to $T$.

The points $A$ and $B$ lie 50 m apart on horizontal ground.

At time $t=0$ two small balls, $P$ and $Q$, are projected in the vertical plane containing $AB$.

Ball $P$ is projected from $A$ with speed 20 m s⁻¹ at 30° to $AB$.

Ball $Q$ is projected from $B$ with speed $u$ m s⁻¹ at angle $\theta$ to $BA$, as shown in Figure 3.

At time $t=2$ seconds, $P$ and $Q$ collide.

Until they collide, the balls are modelled as particles moving freely under gravity.

Find the velocity of $P$ at the instant before it collides with $Q$.

Find

(i) the size of angle $\theta$,

(ii) the value of $u$.

State one limitation of the model, other than air resistance, that could affect the accuracy of your answers.

A particle is projected horizontally from the point with coordinates $(0,18)$ with an initial speed of $12\mathrm{m}{\mathrm{s}}^{-1}$. 

The coordinates are expressed in metres where the $x$ and $y$ coordinates are the horizontal and vertical displacements from the origin respectively.

Find the equation of the trajectory of the particle in the form $y=a+bg{x}^2$ where $a$ and $b$ are constants to be found and $g$ is the acceleration due to gravity.

Find the distance between the particle and the origin after two seconds of motion, giving your answer to three significant figures.

A golfer strikes a ball from ground level with velocity $(20\mathbf{i}+28\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}.$

Find the horizontal distance the golf ball will travel before first hitting the ground.

Show that by reducing the angle of the strike above the horizontal by $10\mathrm{degrees}$ the golfer can achieve approximately 7 m more distance before the ball lands. Assume that the ball is struck at the same speed.

In this question, use $\mathbf{g}\mathbf{=}\mathbf{10}{\mathbf{ms}}^{\mathbf{-}\mathbf{2}}$ for the value of acceleration due to gravity

In a field there are two ball-launchers, $A$ and $B$, aimed at each other.  The ball-launchers are on the same horizontal plane 180 m apart.

Launcher $A$ fires a ball with a velocity of $25\sqrt{3}\mathrm{m}{\mathrm{s}}^{-1}$ at an angle of $\alpha °$ to the horizontal such that $\tan \alpha =\frac{3}{4}$.

At exactly the same moment launcher $B$ launches an identical ball with velocity $(12\sqrt{15}\mathbf{i}+5\sqrt{15}\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.

Assuming that the balls do not collide in mid-air, determine which, if any, of the launchers are struck by a ball. Show your full reasoning and working.

A science-team launches an experimental probe from an underground laboratory toward a research station on the surface 4 km away.

The ground-level is the same at both the station and the laboratory, but because the lab sits deep underground, the probe is launched from a point 600 m below the ground level of the station.

The probe’s initial velocity is $(82.5\mathbf{i}+250\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$, and once released it can be modelled as a projectile moving under gravity alone.

Show that the probe will strike the ground within 1 m of its target and take less than 50 seconds to do so.

The launch shaft which the probe passes through when exiting the laboratory rises 400 m above ground level and can be modelled as a cylinder.

Given that the probe successfully exits the launch shaft, find the minimum possible radius of the shaft.

A simplified diagram of the scenario is shown below.

The flight of a particle projected with an initial velocity of $U\mathrm{m}{\mathrm{s}}^{-1}$ at an angle $\alpha$ above the horizontal is modelled as a projectile moving under gravity only.

The particle is projected from the point  $\left(x_0,y_0\right)$ with the upward direction being taken as positive, and with the coordinates being expressed in metres. $g{\mathrm{m}\mathrm{s}}^{-2}$ is the constant of acceleration due to gravity.

Write down expressions for

(i) the $x$-coordinate of the projectile at time $t$ seconds

(ii) the $y$-coordinate of the projectile at time $t$ seconds.

For a particular projectile, $\tan \alpha =\frac{3}{4}$, $U=10\mathrm{m}{\mathrm{s}}^{-1}$ and the particle is projected from the point $(3,8).$  

Find an expression for the trajectory of the particle, giving your answer in the form

 $y=\frac{a{x}^2+bx+c}{128}$

where the constants $a$, $b$ and $c$ are expressed in terms of $g$.

In a game of “Airwars” one player has to attempt to shoot down another’s toy aircraft in mid-air using a foam missile.  

In a particular game a player launches their aircraft from the origin with velocity $(3\mathbf{i}+18.7\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.  At the same instant their opponent launches their missile with velocity $(-5\mathbf{i}+18.7\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$  from the point with coordinates $(24,0),$ where the coordinates are expressed in metres. 

The flight paths of both the aircraft and the missile occur in the same vertical plane, and i and j  and  are respectively the unit vectors in the positive horizontal and vertical directions (where in the vertical direction upwards is taken to be positive). 

Modelling the motion of both the model aircraft and the model missile as projectiles moving under gravity alone, find the coordinates at which the missile hits the aircraft and how long both had been airborne prior to colliding.