Determine final velocity of projectile motion

1. A kicked football leaves the ground at an angle θ = 30o to the horizontal with an initial speed of 14 m/s. Calculate the final velocity before the ball hits the ground.

Known :

Angle (θ) = 3o

Initial velocity (vo) = 14 m/s

Acceleration of gravity (g) = 10 m/s2

Wanted : Final velocity before the ball hits the ground

Solution :

Solving projectile motion problems - determine final velocity 1Horizontal component of initial velocity :

vox = vo cos θ = (14 m/s)(cos 30o) = (14 m/s)(0.53) = 73 m/s

Vertical component of initial velocity :

voy = vo sin θ = (14 m/s)(sin 30o) = (14 m/s)(0.5) = 7 m/s

Final velocity at vertical direction

Choose upward direction as positive and downward direction as negative.

Known :

Initial velocity (vo) = 7 m/s (positive upward)

Acceleration of gravity (g) = –10 m/s2 (negative downward)

Height (h) = 0 (object back to initial position)

Wanted : Final velocity (vt)

Solution :

vt2 = vo2 + 2 g h = 72 + 2(-10)(0) = 49 – 0 = 49

vt = √49 = 7 m/s

Final velocity at horizontal direction

Initial velocity at horizontal direction is 73 m/s. Velocity is constant so that final velocity is same as initial velocity.

Final velocity before the object hits the ground

Solving projectile motion problems - determine final velocity 2

2. A body is projected upward at an angle of 30o with the horizontal from a building 5 meter high. Its initial speed is 10 m/s. Calculate final velocity before the object hits the ground! Acceleration of gravity is 10 m/s2.

Known :

Angle (θ) = 30o

Initial height (ho) = 5 meters

Initial velocity (vo) = 10 m/s

Acceleration of gravity (g) = 10 m/s2

Wanted : Final velocity

Solution :

Horizontal component of initial velocity :

vox = vo cos θ = (10 m/s)(cos 30o) = (10 m/s)(0.53) = 53 m/s

Vertical component of initial velocity :

voy = vo sin θ = (10 m/s)(sin 30o) = (10 m/s)(0.5) = 5 m/s

Final velocity at vertical direction

Known :

Initial velocity (vo) = 5 m/s (positive upward)

Acceleration of gravity (g) = –10 m/s2 (negative downward)

Height (h) = -5 m (negative because the ground is below the initial height)

Wanted : Final velocity (vt)

Solution :

vt2 = vo2 + 2 g h = 52 + 2(-10)(-5) = 25 + 100 = 125

vt = √125 m/s

Final velocity at horizontal direction

Final velocity at horizontal direction is 5√3 m/s.

Final velocity

Solving projectile motion problems - determine final velocity 3

3. A small ball projected horizontally with initial velocity vo = 8 m/s from a building 12 meters high. Calculate final velocity before ball hits ground! Acceleration of gravity is 10 m/s2

Known :

Height (h) = 12 meters

Initial velocity (vo) = 8 m/s

Acceleration of gravity (g) = 10 m/s2

Wanted : Final velocity (vt)

Solution :

Solving projectile motion problems - determine final velocity 4Horizontal component of initial velocity :

vox = vo = 8 m/s

Vertical component of initial velocity :

voy = 0 m/s

Final velocity at vertical direction

calculated using equation of free fall motion.

Known :

Acceleration of gravity (g) = 10 m/s2

Height (h) = 12 m

Wanted : Final velocity (vt)

Solution :

vt2 = 2 g h = 2(10)(12) = 240

vt = √240 m/s

Final velocity at horizontal direction

Initial velocity at the horizontal direction is 8 m/s. Velocity is constant so that the initial velocity equals the final velocity. So final velocity at horizontal direction is 8 m/s.

Final velocity

Solving projectile motion problems - determine final velocity 5

Read :  Bodies connected by the cord and pulley - application of Newton's law of motion problems and solutions

  1. Resolve initial velocity into horizontal and vertical components
  2. Determine the horizontal displacement
  3. Determine the maximum height
  4. Determine the time interval
  5. Determine the position of the object
  6. Determine the final velocity

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