Determine final velocity of projectile motion

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

Known :

Angle (θ) = 30^o

Initial velocity (v_o) = 14 m/s

Acceleration of gravity (g) = 10 m/s²

Wanted : Final velocity before the ball hits the ground

Solution :

Horizontal component of initial velocity :

v_ox = v_o cos θ = (14 m/s)(cos 30^o) = (14 m/s)(0.5√3) = 7√3 m/s

Vertical component of initial velocity :

v_oy = v_o sin θ = (14 m/s)(sin 30^o) = (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 (v_o) = 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 (v_t)

Solution :

v_t² = v_o² + 2 g h = 7² + 2(-10)(0) = 49 – 0 = 49

v_t = √49 = 7 m/s

Final velocity at horizontal direction

Initial velocity at horizontal direction is 7√3 m/s. Velocity is constant so that final velocity is same as initial velocity.

Final velocity before the object hits the ground

2. A body is projected upward at an angle of 30^o 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) = 30^o

Initial height (h_o) = 5 meters

Initial velocity (v_o) = 10 m/s

Acceleration of gravity (g) = 10 m/s²

Wanted : Final velocity

Solution :

Horizontal component of initial velocity :

v_ox = v_o cos θ = (10 m/s)(cos 30^o) = (10 m/s)(0.5√3) = 5√3 m/s

Vertical component of initial velocity :

v_oy = v_o sin θ = (10 m/s)(sin 30^o) = (10 m/s)(0.5) = 5 m/s

Final velocity at vertical direction

Known :

Initial velocity (v_o) = 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 (v_t)

Solution :

v_t² = v_o² + 2 g h = 5² + 2(-10)(-5) = 25 + 100 = 125

v_t = √125 m/s

Final velocity at horizontal direction

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

Final velocity

3. A small ball projected horizontally with initial velocity v_o = 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 (v_o) = 8 m/s

Acceleration of gravity (g) = 10 m/s²

Wanted : Final velocity (v_t)

Solution :

Horizontal component of initial velocity :

v_ox = v_o = 8 m/s

Vertical component of initial velocity :

v_oy = 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 (v_t)

Solution :

v_t² = 2 g h = 2(10)(12) = 240

v_t = √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

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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