Explanation Given: A projectile when launched at an angle of θ with initial speed v 0 . Proof: Show that the distance travelled is ( v 0 2 / g ) sin 2 θ before hitting the ground. Let the launch point is at the origin, and r ( t ) be the position vector . Let the acceleration due to gravity be g. Then, v ( t ) = ∫ a ( t ) d t = 〈 0 , − g t 〉 + C 0 . ......... (1) The projectile was launched with speed of v 0 at an angle of θ . So, it gives v ( 0 ) = 〈 v 0 cos θ , v 0 sin θ 〉 . From (1), v ( 0 ) = 〈 0 , − g ⋅ 0 〉 + C 0 v ( 0 ) = 〈 0 , 0 〉 + C 0 v ( 0 ) = C 0 C 0 = 〈 v 0 cos θ , v 0 sin θ 〉 Therefore, the equation (1) becomes, v ( t ) = 〈 0 , − g t 〉 + 〈 v 0 cos θ , v 0 sin θ 〉 = 〈 v 0 cos θ , v 0 sin θ − g t 〉 Then, r ( t ) = ∫ 〈 v 0 cos θ , v 0 sin θ − g t 〉 r ( t ) = 〈 v 0 cos θ t , v 0 sin θ t − g 2 t 2 〉 + C 1 At t = 0 , r ( t ) = 〈 0 , 0 〉

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Chapter 13.5, Problem 23E

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To show that a projectile when launched at an angle of θ with initial speed v0 travels a distance (v02/g)sin2θ before hitting the ground. Also, conclude that the maximum distance is attained when θ=π4 for a given v0.

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Chapter 13.5, Problem 22E

Chapter 13.5, Problem 24E

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Chapter 13 Solutions

Calculus: Early Transcendentals

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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To show that a projectile when launched at an angle of θ with initial speed v 0 travels a distance ( v 0 2 / g ) sin 2 θ before hitting the ground. Also, conclude that the maximum distance is attained when θ = π 4 for a given v 0 .

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To show that a projectile when launched at an angle of θ with initial speed v0 travels a distance (v02/g)sin2θ before hitting the ground. Also, conclude that the maximum distance is attained when θ=π4 for a given v0.

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Chapter 13 Solutions