Worksheet: Projectile Motion

In this worksheet, we will practice analyzing the motion of a projectile and finding the parametric equations of motion and the Cartesian equation of its path.

Q1:

A particle projected with a velocity (𝑒+𝑒)ij m/s from a fixed point 𝑂 in a horizontal plane landed at a point in the same plane 360 m away. Find the value of 𝑒 and the projectile’s path’s greatest height β„Ž. Take 𝑔=9.8/ms.

  • A 𝑒 = 4 2 , β„Ž = 1 8 0 m
  • B 𝑒 = 8 4 , β„Ž = 3 6 0 m
  • C 𝑒 = 5 9 . 4 , β„Ž = 1 8 0 m
  • D 𝑒 = 4 2 , β„Ž = 9 0 m
  • E 𝑒 = 5 9 . 4 , β„Ž = 3 6 0 m

Q2:

A particle projected from the origin 𝑂 passed horizontally through a point with a position vector of (10+10)ij m, where i and j are horizontal and vertical unit vectors respectively. Determine the velocity with which the particle left 𝑂, considering the acceleration due to gravity to be 9.8 m/s2.

  • A ( 7 + 2 8 ) i j m/s
  • B ( 1 4 + 2 8 ) i j m/s
  • C ( 7 + 1 4 ) i j m/s
  • D ( 1 4 + 7 ) i j m/s
  • E ( 1 4 + 1 4 ) i j m/s

Q3:

A particle started moving from 𝑂 with a velocity of (2+15)ij m/s, where i and j are unit vectors horizontally and vertically upward respectively. Find, giving your answer to the nearest integer, the distance of the particle from 𝑂 after 1 s. Consider the acceleration due to gravity to be 9.8 m/s2.

Q4:

If a particle was projected from the origin 𝑂 on a horizontal ground with initial velocity (17+11)ij m/s, write its position vector at time 𝑑 seconds. Consider the acceleration due to gravity to be 9.8 m/s2.

  • A  1 7 𝑑 βˆ’ ο€Ή 1 1 𝑑 + 9 . 8 𝑑   i j  m
  • B  1 7 𝑑 + ο€Ή 1 1 𝑑 + 9 . 8 𝑑   i j  m
  • C  1 7 𝑑 + ο€Ή 1 1 𝑑 βˆ’ 9 . 8 𝑑   i j  m
  • D  1 7 𝑑 βˆ’ ο€Ή 1 1 𝑑 + 4 . 9 𝑑   i j  m
  • E  1 7 𝑑 + ο€Ή 1 1 𝑑 βˆ’ 4 . 9 𝑑   i j  m

Q5:

A particle was projected from a point 𝑃 with a position vector (17+9)ij m relative to the origin 𝑂. After 2 seconds, it was at a point (32+50)ij m relative to 𝑂, where i and j are the horizontal and vertical unit vectors respectively. Calculate the distance of the particle from 𝑃 after a further 5 seconds. Take the acceleration due to gravity to be 9.8 m/s2.

Q6:

A man in a balcony threw a ball to his friend, who was standing in the street, horizontally with a speed of 6 m/s. Taking the acceleration due to gravity 9.8 m/s2, find the horizontal and vertical components, v and v, of the ball’s velocity 1 second after it was thrown.

  • A v  = 6 / m s and v=βˆ’4.9/ms
  • B v  = 6 / m s and v=9.8/ms
  • C v  = 6 / m s and v=6/ms
  • D v  = 6 / m s and v=4.9/ms
  • E v  = 6 / m s and v=βˆ’9.8/ms

Q7:

A rock was thrown with a speed of 47 m/s at an angle of elevation 50∘, while a second rock was thrown at an angle of elevation 55∘. If both rocks traveled the same horizontal distance, find, to the nearest two decimal places, the speed with which the second rock was thrown. Take 𝑔=9.8/ms.

Q8:

Find, to one decimal place, a projectile’s speed 3 seconds after it was thrown, given that it flew a total horizontal distance of 282 m for 5 seconds before hitting the ground. Take 𝑔=9.8/ms.

Q9:

Suppose a particle is projected at a speed of 21 m/s and an angle of elevation 51∘. How long will it take for the particle to reach its greatest height? Give your answer correct to two decimal places. Take 𝑔=9.8/ms.

Q10:

As part of a stunt, an aircraft flew up into the sky and then had its engine turned off for a short time. During that time, it was subject to gravitational acceleration of 9.8 m/s2. While the engine was off, the trajectory of the aircraft followed the equation 𝑦=1.77π‘₯βˆ’ο€Ή9.7Γ—10π‘₯οŠͺ. The π‘₯-axis represents horizontal distance, the 𝑦-axis represents vertical distance, and both axes have units in meters. The origin is the point at which the engine was turned off. Find the time taken, in seconds correct to one decimal place, for the aircraft to return to the same height it was at when the engine was turned off.

Q11:

A particle was projected horizontally from a point 42 m above the ground at 32 m/s. Find, to one decimal place, the time it took the particle to reach the ground. Take 𝑔=9.8/ms.

Q12:

A particle was projected from a point 𝑂 in a horizontal plane with a speed of 1.2 m/s and an angle of elevation πœƒ. Determine, in terms of πœƒ how long it takes for the particle to land back on the plane. Take 𝑔=9.8/ms.

  • A 6 πœƒ 4 9 c o s s
  • B 6 πœƒ 4 9 s i n s
  • C 1 2 πœƒ 4 9 s i n s
  • D 2 4 πœƒ 4 9 s i n s
  • E 1 2 πœƒ 4 9 c o s s

Q13:

A particle was projected from the origin 𝑂 with a speed of 16 m/s at an angle of 50∘ above the horizontal. Find, correct to one decimal place, its speed 𝑣 after 2 seconds, and determine the angle πœƒ its direction of travel makes with the horizontal to the nearest degree. Consider the acceleration due to gravity to be 9.8 m/s2.

  • A 𝑣 = 1 6 . 0 / m s , πœƒ = 3 7 ∘
  • B 𝑣 = 1 2 . 6 / m s , πœƒ = 3 6 ∘
  • C 𝑣 = 1 6 . 0 / m s , πœƒ = 7 2 ∘
  • D 𝑣 = 1 2 . 6 / m s , πœƒ = 5 0 ∘
  • E 𝑣 = 2 8 . 8 / m s , πœƒ = 6 9 ∘

Q14:

Elizabeth threw a ball at an angle of elevation of 𝛼 to the horizontal, where tan𝛼=0.44. Her brother Daniel, who was 3.8 m away, caught it at the same height that it was thrown. Taking 𝑔=9.8/ms, determine the speed with which the ball was thrown correct to two decimal places.

Q15:

A particle is projected from a point on the ground and moves freely under gravity. The particle’s height above the ground, 𝑦 meters, is related to its horizontal distance from the point of projection, π‘₯ meters, by the equation 𝑦=0.27π‘₯βˆ’ο€Ή5.4Γ—10π‘₯οŠͺ. Taking 𝑔=9.8/ms, find the speed and angle of elevation at which the particle was projected. Give these values correct to one decimal place.

  • A 139.5 m/s, 1 5 . 1 ∘
  • B 292.2 m/s, 7 4 . 9 ∘
  • C 517.1 m/s, 7 4 . 9 ∘
  • D 98.7 m/s, 1 5 . 1 ∘
  • E 365.7 m/s, 7 4 . 9 ∘

Q16:

A particle was projected from a point 𝑂 on horizontal ground. If, after 8 seconds, its horizontal distance from 𝑂 was 48 m and its height was 98 m, determine how far the particle would be from 𝑂 when it hits the ground. Take 𝑔=9.8/ms.

Q17:

William threw a ball from the top of a building at a speed of 9 m/s toward his brother, who was standing 13 m away from the building. If the ball’s initial direction was 25∘ below the horizontal, determine its speed 𝑣 and direction πœƒ when his brother caught it correct to two decimal places. Take 𝑔=9.8/ms.

  • A 𝑣 = 8 . 1 6 / m s , πœƒ = 6 8 . 9 3 ∘ below the horizontal
  • B 𝑣 = 2 1 . 0 7 / m s , πœƒ = 6 8 . 9 3 ∘ below the horizontal
  • C 𝑣 = 2 1 . 0 7 / m s , πœƒ = 6 7 . 2 2 ∘ below the horizontal
  • D 𝑣 = 1 9 . 4 2 / m s , πœƒ = 6 7 . 2 2 ∘ below the horizontal
  • E 𝑣 = 8 . 1 6 / m s , πœƒ = 2 3 . 9 3 ∘ below the horizontal

Q18:

A particle is projected with a speed of 23 m/s at an angle of elevation πœƒ. It reached a maximum height of 17 m above where it was projected. Taking 𝑔=9.8/ms, find πœƒ to the nearest degree.

Q19:

A flare was fired from a flare gun. After 2 seconds, the flare’s horizontal and vertical distances from the flare gun were 22 m and 31 m respectively. Find the horizontal v and vertical v components of the flare’s initial velocity. Take 𝑔=9.8/ms.

  • A v  = 1 1 m/s, v  = 5 . 7 m/s
  • B v  = 1 1 m/s, v  = 2 5 . 3 m/s
  • C v  = 5 . 5 m/s, v  = 2 5 . 3 m/s
  • D v  = 2 2 m/s, v  = 2 5 . 3 m/s
  • E v  = 5 . 5 m/s, v  = 5 . 7 m/s

Q20:

A stone is thrown toward a wall from a point 𝑂 on horizontal ground with a speed of 25 m/s at an angle of 40∘ above the horizontal. If the wall is 39 m away from 𝑂, how fast is the stone when it hits it? Give your answer correct to one decimal place. Take 𝑔=9.8/ms.

Q21:

A particle is projected horizontally from a point 11 m above the ground at 14 m/s. Find the distance between its point of projection and the point where it hit the ground to the nearest meter. Take 𝑔=9.8/ms.

Q22:

A particle was projected from the origin 𝑂 with a speed of 30 m/s at an angle of 36∘ above the horizontal. Find, correct to one decimal place, its speed after 4 seconds, and determine the angle πœƒ its direction of travel makes with the horizontal to the nearest degree. Consider the acceleration due to gravity to be 9.8 m/s2.

  • A 𝑣 = 3 2 . 5 / m s , πœƒ = 4 2 ∘
  • B 𝑣 = 3 2 . 5 / m s , πœƒ = 3 6 ∘
  • C 𝑣 = 6 5 . 4 / m s , πœƒ = 6 8 ∘
  • D 𝑣 = 6 5 . 4 / m s , πœƒ = 6 7 ∘
  • E 𝑣 = 3 0 . 0 / m s , πœƒ = 4 0 ∘

Q23:

A rock was thrown from the top of a building at a speed of 11 m/s and an angle of elevation 𝛼, where tan𝛼=43. Given that it took 3 s for the rock to hit the ground, find the horizontal distance the rock traveled.

Q24:

Two particles, 𝑃 and 𝑄, are projected simultaneously, moving freely under gravity: particle 𝑄 is projected from the ground at 28.6 m/s at an angle πœƒ above the horizontal, and particle 𝑃 is projected horizontally at 12.9 m/s from a point 88 m above 𝑄’s point of projection. If both particles collided in the air, find πœƒ and determine how long it takes for the particles to collide from the moment they were both projected. Give your answers to one decimal place, taking 𝑔=9.8/ms.

  • A πœƒ = 6 3 . 2 ∘ , 𝑑 = 6 . 8 s
  • B πœƒ = 2 6 . 8 ∘ , 𝑑 = 3 . 4 s
  • C πœƒ = 2 6 . 8 ∘ , 𝑑 = 6 . 8 s
  • D πœƒ = 6 3 . 2 ∘ , 𝑑 = 3 . 4 s
  • E πœƒ = 2 4 . 3 ∘ , 𝑑 = 7 . 5 s

Q25:

An archer shot an arrow with a speed of 80 m/s at an angle of 20∘ above the horizontal. Calculate the arrow’s horizontal range π‘₯. Also calculate its vertical height β„Ž above the launch point when it was at a distance of 46 m from the archer. Give your answers in meters correct to one decimal place. Consider the acceleration due to gravity to be 9.8 m/s2.

  • A π‘₯ = 4 1 9 . 8 m , β„Ž = 1 1 2 . 5 m
  • B π‘₯ = 8 3 9 . 6 m , β„Ž = 4 2 . 3 m
  • C π‘₯ = 2 0 9 . 9 m , β„Ž = 1 1 2 . 5 m
  • D π‘₯ = 2 0 9 . 9 m , β„Ž = 1 4 . 9 m
  • E π‘₯ = 4 1 9 . 8 m , β„Ž = 1 4 . 9 m

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