Question

You fire an arrow at an angle of 45o from the horizontal and it lands a distance of 100 m from you. If you fire an identical arrow at the same speed but at 46o, how far away from you will it land? Ignore air resistance.

Answer 1

Firing the arrow at a 46-degree angle will result in a distance very close to 100 meters, slightly less due to the increased vertical component of the initial velocity.

Step-by-step explanation:

The trajectory and range of a projectile launched under gravity at an angle are governed by the equations of motion for projectile motion. When an arrow is fired at a 45° angle and hits the ground 100 m away, its initial speed can be calculated using the range formula for projectile motion. However, if the angle is changed to 46° while the initial speed remains the same, the range will change slightly due to the dynamics dictated by the motion's angle.

For small changes in launch angle around the optimal angle of 45°, the change in range is minimal because the function of range versus angle reaches a maximum around 45°. Therefore, firing an arrow at 46° will land at a distance that is very close to 100 m, possibly slightly less due to the increase in the vertical component of the initial velocity and the symmetry of the projectile motion trajectory around 45°.

Answer 2

Arrow projected at an angle of 46⁰ will travel 99.94 m.

Explanation:

The arrow fired at an angle 45⁰ travels 100 m horizontally, the horizontal movement is because of horizontal component of velocity.

Horizontal component = u cos θ, where u is the velocity of projection.

The time taken to travel 100m is depending on vertical component of velocity, time taken is calculated by dividing vertical component of velocity by acceleration due to gravity and multiplying by two.

So, time taken = 2u sin θ/g

Horizontal distance travelled by projectile,
`H = u cos \theta * (2u sin \theta )/(g) = (2u^2 sin 2\theta )/(g)`

So. H ∝ sin 2θ

Here
`(H_1)/(H_2) = (sin 2\theta_1 )/(sin 2\theta_2)\\ \\ (100)/(H_2) = (sin (2*45) )/(sin (2*46))=(1)/(sin 92) \\ \\ H_2=99.94 m`

So an arrow projected at an angle of 46⁰ will travel 99.94 m.