Mechanics

15 methods · 77 questions

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For unpowered flight, motion is uniform horizontally and uniformly accelerated vertically. Write $x = u t cos θ$ and $y = u t sin θ - \frac{1}{2} g t^{2}$ independently, then eliminate $t$ to get the trajectory $y = x tan θ - \frac{g x ^{2}}{2 u ^{2}} sec^{2} θ$ , or impose a meeting/clearance condition. Many "show that" relations drop out by reading off times of flight, maximum height $u^{2} sin^{2} θ /2 g$ and range.

Trigger: any projectile fired with speed and angle, especially "just clears", "passes over both walls", "hits the stumps" or "collide in the air".

Instances:
(i) impose $y = h$ at a given $x$ to clear an obstacle;
(ii) equate two particles' $x$ and $y$ at the same $t$ for a mid-air collision;
(iii) treat $tan θ$ as the unknown and read off a quadratic in it.

Linked questions (3)

STEP 2 2006 · Q11

A projectile of unit mass is fired in a northerly direction from a point on a horizontal plain at speed

u

and an angle

θ

above the horizontal. It lands at a point

A

on the plain. In flight, the projectile experiences two forces: gravity, of magnitude

g

; and a horizontal force of constant magnitude

f

due to a wind blowing from North to South. Derive an expression, in terms of

u

g

f

and

θ

for the distance

O A

Determine the angle $α$ such that, for all $θ > α$ , the wind starts to blow the projectile back towards $O$ before it lands at $A$ .
An identical projectile, which experiences the same forces, is fired from $O$ in a northerly direction at speed $u$ and angle $4 5^{\circ}$ above the horizontal and lands at a point $B$ on the plain. Given that $θ$ is chosen to maximise $O A$ , show that $\frac{O B}{O A} = \frac{g - f}{g ^{2} + f ^{2} - f} .$ Describe carefully the motion of the second projectile when $f = g$ .

STEP 2 2008 · Q9

In this question, use

g = 10 m s^{- 2}

.

In cricket, a fast bowler projects a ball at

40 m s^{- 1}

from a point

h m

above the ground, which is horizontal, and at an angle

α

above the horizontal. The trajectory is such that the ball will strike the stumps at ground level a horizontal distance of

20 m

from the point of projection.

Determine, in terms of $h$ , the two possible values of $tan α$ . Explain which of these two values is the more appropriate one, and deduce that the ball hits the stumps after approximately half a second.
State the range of values of $h$ for which the bowler projects the ball below the horizontal.
In the case $h = 2.5$ , give an approximate value in degrees, correct to two significant figures, for $α$ . You need not justify the accuracy of your approximation.

[You may use the small-angle approximations

cos θ \approx 1

and

sin θ \approx θ

STEP 3 2009 · Q9

A particle is projected under gravity from a point

P

and passes through a point

Q

. The angles of the trajectory with the positive horizontal direction at

P

and at

Q

are

θ

and

ϕ

, respectively. The angle of elevation of

Q

from

P

α

Show that $tan θ + tan ϕ = 2 tan α$ .
It is given that there is a second trajectory from $P$ to $Q$ with the same speed of projection. The angles of this trajectory with the positive horizontal direction at $P$ and at $Q$ are $θ^{'}$ and $ϕ^{'}$ , respectively. By considering a quadratic equation satisfied by $tan θ$ , show that $tan (θ + θ^{'}) = - cot α$ . Show also that $θ + θ^{'} = π + ϕ + ϕ^{'}$ .