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EXAMPLE 4

Soccer. Write an equation for the horizontal distance traveled by a soccer ball kicked from ground level ( h 0 = 0) at speed v and angle q. EXAMPLE 4. Derive a trigonometric model. 16. –. = 0. x 2 + ( tan q ) x + 0. v 2 cos 2 q. 1. 16. 16. = 0. – x (. x – tan q ).

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EXAMPLE 4

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  1. Soccer Write an equation for the horizontal distance traveled by a soccer ball kicked from ground level (h0 = 0) at speed vand angle q. EXAMPLE 4 Derive a trigonometric model

  2. 16 – = 0 x2 + (tanq )x + 0 v 2 cos2q 1 16 16 = 0 – x( x – tanq ) v 2 cos2q 16 = 0 x – tanq v 2 cos2q 16 = tanq x v 2 cos2q Multiply each side by 1 x v2 cos2q. 16 = v2 cos2q tan q EXAMPLE 4 Derive a trigonometric model SOLUTION Leth0 = 0. Factor. Zero product property Add tan q to each side.

  3. = x v2 cos qsinq 1 1 1 1 1 x = v2(2cos qsinq) Rewrite as 2. 16 32 16 32 32 x = v2sin 2q EXAMPLE 4 Derive a trigonometric model Use cos q tan q= sin q. Use a double-angle formula.

  4. REASONING: Use the equation x = v2sin 2 q to explain why the projection angle that maximizes the distance a soccer ball travels is q = 45°. 10. 1 32 ANSWER ANSWER No sin 2(45°) = sin 90° = 1, this is the only value that will not decrease the velocity of the ball since it is the only non-fractional value. for Examples 4 GUIDED PRACTICE 9. WHAT IF? Suppose you kick a soccer ball from ground level with an initial speed of 70 feet per second. Can you make the ball travel 200 feet?

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