Question

Falling-Body Problems Use the formula  discussed in Example 7.

A ball is thrown straight upward at an initial speed of v₀ = 40 ft/s.

(a) When does the ball reach a height of 24 ft?

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(b) When does it reach a height of 48 ft?

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(c) What is the greatest height reached by the ball?

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(d) When does the ball reach the highest point of its path?

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(e) When does the ball hit the ground?

Answer 1

Since in this case the initial speed the formula is

(a) Setting gives the equation

[set ]

[all terms to LHS]

[divide by 8]

[factor]

[solve]

The ball reaches 24ft in 1s (on its ascent) and again after 1.5s (on its descent to Earth)

(b) Setting gives the equation

[set ]

[all terms to LHS]

[divide by 8]

The discriminate of this equation

Which is negative.

Thus, the equation has no real solution

The ball never reaches a height of 49ft

(c) Each height the ball reaches is attained twice, once on its ascent and once on its descent. The only exception is the highest point of its path, which is reached only once. This means that for the highest value of h, the following equation has only one solution for t.

[all terms to LHS]

This is turn means that the discriminate D of the equation is 0, so

The maximum height reached is 25ft

(e) The ground level corresponds to , so we must solve the equation

[set ]

[factor]

Thus, or

This means that the ball starts at ground level and returns to ground level after