Question

Four bodies are projected with
the same velocities at angles
30°, 40°, 55 and 65° with the
horizontal. The horizontal
range will be largest for
the
one projected at an angle
A
30
B
40
C
55°
D 65°

Answer 1

Answer

Formula

Range of a projectile motion is given by:

R = $U^2$ sin(2θ) / g

• R is the range of projectile
• θ is the angle at which it is projected
• U is initial velocity
• g is gravitational acceleration

So, we can say the range is directly proportional to sin(2θ)

R ∝ sin(2θ)

Method-1

Using direct values of sin(2 x 30) , sin(2 x 40), sin(2 x 55) and sin(2 x 65)

sin (60) = 0.866

sin (80) = 0.98

sin (110) = 0.93

sin (130) = 0.76

So, The horizontal range will be largest for the one projected at an angle 40°

Method-2

Without using direct values

Concept to be used-

Range projected at angle θ and at angle 90-θ will be same.

So, we can say that range of the body projected at 55° will be equal to (90-55)° = 35°

Range of the body projected at 65° will be equal to (90-65)° = 25°

Since the range is directly proportional to twice of θ,

2 x 30° = 60°

2 x 40° = 80°

2 x 35° = 70°

2 x 25° = 50°

Value of sin x always increases from 0 to 90°

Maximum value of 2θ is 80°.

So, The horizontal range will be largest for the one projected at an angle 40°

Answer 2

The horizontal  range will be largest for  the  one projected at an angle D. $65^{o}$.

Explanation:

• When any object is thrown obliquely in air, the only force acting on it is the force due to gravity.
• The body moves along a curved path with constant acceleration. This acceleration is directed towards the centre of earth. This curved trajectory motion is called projectile motion.

There are two types of rectilinear motion in projectile motion:

1. Along x-axis the body has horizontal motion due to horizontal component of gravitational force. Its motion is with uniform velocity.
2. Along y-axis the body has downward motion due to vertical component of gravitational force. Its motion is with uniform acceleration.

There are three main parameters in projectile which are expressed as:

Time of flight (T):

• It is the time duration for the body to complete its motion.

• $T=\frac{2usin\alpha }{g}$, where α is the angle of projection, g is the gravitational acceleration and $u$ is the initial velocity.

Horizontal Range (R):

• It is the horizontal displacement of the object after its motion.

• $R=\frac{u^{2} sin 2\alpha }{g}$, where α is the angle of projection, g is the gravitational acceleration and $u$ is the initial velocity.

Maximum Height (H):

• It is the maximum vertical distance covered by the object.

• $H =\frac{u^{2} sin^{2}\alpha }{2g}$, where α is the angle of projection, g is the gravitational acceleration and $u$ is the initial velocity.

Since Range of the projectile motion is directly proportional to sine value of angle of projection when velocities are same for all projections. So, Range is maximum for the maximum sine value of angle.

Therefore, range is maximum for $65^{o}$.