Projectile motion – problems and solutions

1. A bullet fired at an angle θ = 60o with a velocity of 20 m/s. Acceleration due to gravity is 10 m/s2. What is the time interval to reach the maximum height?

Known :

The initial velocity of bullet (vo) = 20 m/s

Angle (θ) = 60oC

Acceleration due to gravity (g) = 10 m s–2

Wanted : The time interval to reach the maximum height

Solution :

The initial velocity at the horizontal direction (x axis) :

vox = vo cos 60o = (20)(0.5) = 10 m/s

The initial velocity at the vertical direction (y axis) :

voy = vo sin 60o = (20)(0.5√3) = 10√3 m/s

The time interval to reach the maximum height, calculated using this equation :

vty = voy + g t

vty = the final velocity in the vertical direction = the final velocity at the highest point = 0 m/s

voy = the initial velocity at the horizontal direction = 10√3 m/s

g = acceleration due to gravity = 10 m/s2

t = time interval

The time interval :

vty = voy + g t

0 = 10√3 – 10 t

10√3 = 10 t

t = 10√3 / 10

t = √3 seconds

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2. An object projected at an angle. The height of the object is the same when the time interval = 1 second and 3 seconds. What is the time interval the object in air.

Solution :

Projectile motion – problems and solutions 1

The object in the air for 4 seconds.

3. An aircraft is moving horizontally with a speed of 50 m/s. At the height of 2 km, an object is dropped from the aircraft. Acceleration due to gravity = 10 m/s2, what is the time interval before the object hits the ground.

Known :

Height = 2 km = 2000 metersProjectile motion – problems and solutions 2

Acceleration due to gravity (g) = 10 m/s2

Wanted : The time interval (t)

Solution :

h = 1/2 g t2

2000 = 1/2 (10) t2

2000 = 5 t2

t2 = 2000/5 = 400

t = 400 = 20 seconds

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4. A kicked football leaves the ground at an angle θ = 45o with the horizontal has an initial speed of 25 m/s. Determine the distance of X. Acceleration due to gravity is 10 m/s2.

Known :Projectile motion – problems and solutions 5

Initial speed (vo) = 25 m/s

Acceleration due to gravity (g) = 10 m/s2

Angle (θ) = 45o

Wanted : X

Solution :

The horizontal component of the initial velocity :

vox = vo cos θ = (25 m/s)(cos 45o) = (25 m/s)(0.52) = 12.52 m/s

The vertical component of the initial velocity :

voy = vo sin θ = (25 m/s)(sin 45o) = (25 m/s)(0.52) = 12.52 m/s

Projectile motion could be understood by analyzing the horizontal and vertical component of the motion separately. The x motion occurs at constant velocity and the y motion occurs at constant acceleration of gravity.

Time in the air (t) :

The time in air calculated with the equation of the upward vertical motion.

Choose upward direction as positive and downward direction as negative.

Known :

The initial velocity (vo) = 12.52 m/s (upward direction, positive)

Acceleration due to gravity (g) = -10 m/s2 (downward direction, negative)

Height (h) = 0

Wanted : Time interval (t)

Solution :

h = vo t + 1/2 g t2

0 = (12.52) t + 1/2 (-10) t2

0 = 12.52 t – 5 t2

12.52 t = 5 t2

12.52 = 5 t

t = 12.52 / 5

t = 2.52 seconds

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The horizontal distance (X) :

Calculated using the equation of the uniform linear motion with constant velocity.

Known :

Velocity (v) = 12.52 m/s

Time interval (t) = 2.52 seconds

Wanted : Distance

Solution :

d = v t = (12.52)(2.52) = (12.5)(2.5)(2) = 62.5 meters

5. An object projected upward at an angle θ = 30o with the horizontal has an initial speed of 20 m/s. Acceleration due to gravity is 10 m/s2. Determine the maximum height.

Known :

The initial velocity (vo) = 20 m/sProjectile motion – problems and solutions 6

Acceleration due to gravity (g) = 10 m/s2

Angle (θ) = 30o

Wanted : The maximum height

Solution :

First, find the vertical component of the initial velocity (voy) :

voy = vo sin 30o = (20)(sin 30o) = (20)(0.5) = 10 m/s

Calculate the maximum height. Choose upward direction as positive and downward direction as negative.

Known :

Acceleration due to gravity (g) = -10 m/s2 (downward direction, negative)

The vertical component of the initial velocity (voy) = 10 m/s (upward direction, positive)

Velocity at the maximum height (vty) = 0

Wanted : The maximum height (h)

Solution :

vt2 = vo2 + 2 g h

02 = 102 + 2 (-10) h

0 = 100 – 20 h

100 = 20 h

h = 100/20

h = 5 meters

The maximum height is 5 meters.

6. An object is thrown at a certain elevation angle. The height of the object same after 1 second and 3 seconds. Determine time in air.

A. 3.6 sProjectile motion – problems and solutions 1

B. 4.0 s

C. 5.6 s

D. 6.4 s

Solution

Time in air = 4 seconds.

The correct answer is B.

7. An aircraft is moving horizontally with the speed of 50 m/s. When the aircraft at the height of 2 km, an object free fall from the aircraft. Determine the type of the motion.

A. Free fall motionProjectile motion – problems and solutions 2

B. Floating motion

C. Horizontal motion

D. Projectile motion

Solution :

The object is dropped from the moving plane because it has the same speed as the plane’s speed, that is 50 m/s. Movement of objects is not like free fall motion but parabolic motion. The case is the same as you are dropping objects from inside a moving car.

The correct answer is D.