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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Shot-Put Throw The path of a shot thrown at an angle θ is

r ( t ) = ( v 0 cos θ ) t i + h + ( v 0 sin θ ) t 1 2 g t 2 j

where v 0 is the initial speed, h is the initial height, t is the time in seconds, and g is the acceleration due to gravity. Verify that the shot will remain in the air for a total of

t = v 0 sin θ + v 0 2 sin 2 θ + 2 g h g seconds

and will travel a horizontal distance of

v 0 2 cos θ g ( sin θ + sin 2 θ + 2 g h v 0 2 ) feet.

To determine

To Prove: The shot will remain in the air for a total of t=v0sinθ+v02sin2θ+2ghg seconds and will travel a horizontal distance of v02cosθg(sinθ+sin2θ+2ghv02) feet if the path of a shot thrown at an angle θ is r(t)=(v0vosθ)ti+[h+(v0sinθ)t12gt2]j.

Explanation

Given:

Time is t=v0sinθ+v02sin2θ+2ghg seconds.

Horizontal distance is v02cosθg(sinθ+sin2θ+2ghv02) feet.

Path of a shot is r(t)=(v0vosθ)ti+[h+(v0sinθ)t12gt2]j.

Formula used:

The quadratic formula is:

x=b±b24ac2a.

Proof:

Consider the position vector of the projectile,

r(t)=(v0vosθ)ti+[h+(v0sinθ)t12gt2]j

To find the range setting substitute y=0,

y=h+(v0sinθ)t12gt2=0

Therefore by applying quadratic formula,

(12g)t2(v0sinθ)th=0t=v0sinθ

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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