Difficult Projectile Motion Questions

Some very difficult extension questions on projectile motion, I'll be very impressed if anyone can get these:

1) A golf ball is hit at 60ms^-1. At what angle should it leave the club in order to travel 250m horizontally? You will need to use the double angle formula

sin2A = 2 sinA cosA

.

2) Prove that, in the absence of air resistance, the maximum range of any projectile is achieved when it is launched at 45^o to the horizontal.

3) How can two projectiles launched with the same speed but at different angles have he same range?

Reply 1

rbnphlp

11

Original post by Ivo

Some very difficult extension questions on projectile motion, I'll be very impressed if anyone can get these:

1) A golf ball is hit at 60ms^-1. At what angle should it leave the club in order to travel 250m horizontally? You will need to use the double angle formula

sin2A = 2 sinA cosA

.

2) Prove that, in the absence of air resistance, the maximum range of any projectile is achieved when it is launched at 45^o to the horizontal.

3) How can two projectiles launched with the same speed but at different angles have he same range?

for the 1st one start with , resolving velocities into 60cosx , and 60sinx ..

and consider motion in vertical plane find the time taken

use this time and horizontal distance and speed (60cosx) to find x.

Its the same kind of thing for 2nd one as well

Reply 2

BrilliantMinds

2) R = u^2 sin2x / g. For maximum Range R, sin 2x = ?

Equate both R's:

u^2 sin 2x / g = u^2 sin2y/g

sin 2x = sin 2y

(Weird case: imagine if x = 90 deg, y = 0 both will have 0 range.)

Reply 3

Desk-Lamp

1

Part 2 was used in my uni interview :smile:

Personally I think the questions are in the wrong order, once you get the distance as a function of angle and velocity for part 2 questions 1/3 become fairly trivial.

Reply 4

Curiouslad

The question asks us to figure out the initial velocity (v) and also the time of the flight.
I have made two equations:
v = velocity, t = total time of flight,
1. v cos(theta) * t = 20
2. 3 = v sin(theta) * t + ((10 t^ 2)/2)
Any help?

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Reply 5

GeddyBaby

4

The time of flight is given in this question as 0.330 seconds - should make it solvable now.

Reply 6

महादेव

1

For max range angle=45°Sin2A=1Sin2A=90°SinA=45°

Reply 7

DAVIDDIVINE

2

Original post by Ivo

Some very difficult extension questions on projectile motion, I'll be very impressed if anyone can get these:

1) A golf ball is hit at 60ms^-1. At what angle should it leave the club in order to travel 250m horizontally? You will need to use the double angle formula

sin2A = 2 sinA cosA

.

2) Prove that, in the absence of air resistance, the maximum range of any projectile is achieved when it is launched at 45 ^o to the horizontal.

3) How can two projectiles launched with the same speed but at different angles have he same range?

THE FIRST QUESTION'S ANSWER

Range =U^2 sin 2A/ g Take g = 10 ms-2
250 = 60^2 sin 2A/ 10
250*10 = 3600 sin 2A
2500/3600 = sin 2A
0.6944/ 2 = sin A
0.3472 = sin A
0.3472*sin-1 = A
A = 20.31

THE SECOND QUESTION'S ANSWER

The maximum height occurs when the vertical speed - Vy = 0
V0y = V0sinθ
As a function of time, t: Vy = V0sinθ−gt
When Vy = 0:
(1):V0sinθ = gt
Now horizontal displacement: x = V0xt = V0cosθt

(2):t = xV0cosθ

Substituting for t from (2) in (1): V0sinθ = gx/ V0cosθ
=2(V0)^2cosθsinθg

The range is twice this distance so: xmax=2(V0)^2cosθsinθ/g

Substituting the trigonometric identity:2sinθcosθ=sin2θ gives:

xmax=(V0)^2sin2θ/g

Since sin2θ is maximum at 2θ=π/2 then maximum horizontal displacement occurs at θ=π/4
This proves that the maximum range is achieved when it is launched at 45 degree angle

THE THIRD QUESTION

This can happen if the angles at which the objects are launched at are complementary
let me explain this with an equation

the range is proportional to sinAcosA = sin 2A

this is shown as

sin(180−A)=sinA

sin2A=sin2(90−A)

So the range of the projectiles is directly proportional to the sine of twice the angle and if everything else was constant the range will be equal when

sin (2A) = sin (2B) ( where A and B are different angles)

For the above equation to be true :-

Either 2A = 2B

- A = B

or

2A = 180- 2B

=> A = 90 - B