Determine the maximum height of projectile motion

Solved problems in projectile motion – determine the maximum height

1. A kicked football leaves the ground at an angle θ = 60^o with the horizontal has an initial speed of 10 m/s. Calculate the maximum height! Acceleration of gravity is 10 m/s².

Known :

Angle (θ) = 60^o

Initial speed (v_o) = 10 m/s

Wanted : Maximum height (h)

Solution :

Vertical component of initial velocity :

sin 60^o = v_oy / v_o

v_oy = v_o sin 60^o = (10)(sin 60^o) = (10)(0.5√3) = 5√3 m/s

Choose upward direction as positive and downward direction as negative.

Known :

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

Vertical component of initial velocity (v_oy) = +5√3 m/s (positive upward)

Final velocity at the maximum height (v_ty) = 0

Wanted : Maximum height (h)

Solution :

v_t² = v_o² + 2 g h

0² = (5√3)² + 2 (-10) h

0 = 25(3) – 20 h

0 = 75 – 20 h

75 = 20 h

h = 75 / 20

h = 3.75 meter

The maximum height is 3.75 meter.

2. A body is projected upward at angle of 30^o with the horizontal from a building 20 meter high. It’s initial speed is 4 m/s. Calculate the maximum height! Acceleration of gravity is 10 m/s².

Known :

Angle (θ) = 30^o

Initial height (h) = 20 meter

Initial velocity (v_o) = 4 m/s

Acceleration of gravity (g) = 10 m/s²

Wanted : The maximum height (h)

Solution :

Vertical component of initial velocity :

sin 30^o = v_oy / v_o

v_oy = v_o sin 30^o = (4)(sin 30^o) = (4)(0.5) = 2 m/s

Choose upward direction as positive and downward direction as negative.

Known :

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

Vertical component of initial velocity (v_oy) = +2 m/s (positive upward)

Final velocity at maximum height (v_ty) = 0

Wanted : The maximum height

Solution :

The maximum height :

v_t² = v_o² + 2 g h

0² = 2² + 2 (-10) h

0 = 4 – 20 h

4 = 20 h

h = 4 / 20

h = 0.2 meter

The maximum height is 0.2 meter + 20 meter = 20.2 meter.

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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 object
6. Determine the final velocity