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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

If a projectile is fired with an initial velocity of v0 meters per second at an angle α above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations

x = ( v 0 cos α ) t y = ( v 0 sin α ) t 1 2 g t 2

where g is the acceleration due to gravity (9.8 m/s2).

  1. (a) If a gun is fired with α = 30° and v0 = 500 m/s, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet?
  2. (b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several other values of the angle α to see where it hits the ground. Summarize your findings.
  3. (c) Show that the path is parabolic by eliminating the parameter.

(a)

To determine

To find: The maximum height and distance reached by the bullet.

Explanation

Given data:

The parametric equation for the variable x is as follows.

x=(v0cosα)t (1)

The parametric equation for the variable y is as follows.

y=(v0sinα)t12gt2 (2)

Calculation:

Substitute 500m/s for v0, π6 for α and 9.8m/s2 for g in Equation (1).

x=(500cos(π6))t=50032t=2503t

Substitute 500m/s for v0 and π6 for α in Equation (2).

y=[500sin(π6)]t12gt2=5002t12gt2=250t12gt2 (3)

The bullet will hit the ground when y=0

0=250t12gt2t=250×2g

Substitute 9.8m/s2 for g in the above equation.

t=500m/s9

(b)

To determine

To find: The path of the projectile for different values of the angle α and to find where bullet hits the ground.

(c)

To determine

To find: The parabolic equation for the given parametric equation.

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