a soccer ball is kicked at an angle of 35° and it lands on even ground.
A) What angle will produce the same range?
B) Which angle of the two will produce the highest ball?
C) Which angle of the two will produce the ball that is in the air the longest?
D) What angle in this situation would produce the furthest range?

Answers

Answer 1

Hi there!

A)

The angle that will produce the same range is the compliment of 35°.

Thus, kicking the ball at 55° will result in the same range.

We can prove this by using the derived range equation:

[tex]R = \frac{v^2sin2\theta}{g}[/tex]

An angle of 35° yields:

[tex]R = \frac{v^2sin(2*35)}{g} = .939R[/tex]

An angle of 55° yields:

[tex]R = \frac{v^2sin(2*55)}{g} = .939R[/tex]

Both are the same, thus indicating that 55° produces the same range.

B)

The angle of 55° will produce the higher ball because the VERTICAL component of the ball's velocity is greater compared to kicking the ball at 35° degree.

sin(55) > sin(35)

C)

The angle of 55° will result in the ball being in the air the longest because when a ball is in the air (assuming no air resistance), the ball experiences an acceleration due to gravity of -9.8 m/s², causing the vertical velocity to decrease until it eventually reaches 0 m/s at the top of its path. A greater initial vertical velocity means that it will take longer for the ball to fall.

We can prove this using:

vf = vi + at

0 = vy - 9.8t

vy/9.8 = t

Greater vy (vertical component of velocity) ⇒ greater time taken.

D)

The angle that would result in the furthest range is 45°.

We can prove this using calculus. Recall the above range equation:

[tex]R = \frac{v^2sin2\theta}{g}[/tex]

We can take the derivative and use the first-derivative test to find its critical point:

[tex]\frac{dR}{d\theta} = \frac{v^22cos2\theta}{g} = 0[/tex]

Evaluate:

[tex]v^22cos2\theta = 0 \\\\cos2\theta = 0 \\\\2\theta = 90^o\\\\\boxed{\theta = 45^o}[/tex]

Answer 2

A. The angle that will produce the same range as a 35° kick is 53°.

B. The angle that produces the highest ball is 45°.

C. The angle that produces the ball that is in the air the longest is 45°.

D. There is no single angle that will produce the furthest range for all initial velocities and accelerations due to gravity.

A. The range of a projectile is the horizontal distance it travels before it hits the ground. The range of a projectile is determined by the initial velocity, the angle of projection, and the acceleration due to gravity.

For a given initial velocity, the range of a projectile is maximized when the angle of projection is 45°. However, if the ground is not level, the range of a projectile can be maximized at other angles.

In the case of a soccer ball kicked at an angle of 35°, the range will be maximized at an angle of 53°. This is because the range of a projectile is maximized when the vertical component of the initial velocity is equal to the horizontal component of the initial velocity.

The angle of 53° is the angle that produces a vertical component of the initial velocity that is equal to the horizontal component of the initial velocity when the ball is kicked at an angle of 35°.

Therefore, the angle that will produce the same range as a 35° kick is 53°.

B. The angle that produces the highest ball is 45°.

C. The angle that produces the ball that is in the air the longest is 45°.

D. The angle that produces the furthest range depends on the initial velocity and the acceleration due to gravity. There is no single angle that will produce the furthest range for all initial velocities and accelerations due to gravity.

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Image credit: The Hubble Heritage Team (AURASTSCI/NASA)
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The attached picture explains the law of conservation of energy.

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