CritABCD
A.1 · 9b · A.1 Kinematics · projectile motion, independence of horizontal and vertical components

Independence of motion: x and y

Drop one ball, throw another sideways. They land together. Horizontal and vertical motion do not depend on each other, and only the time links them.

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Hook

Two balls leave the same height at the same moment. One is simply dropped. The other is thrown sideways. Which hits the ground first? Your gut says the dropped one. Your gut is wrong.

The question
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Ball A is dropped. Ball B is thrown sideways.

50 mt = T / 3t = 2T / 3AB5 m s⁻¹ABR = ? (Q2)displacement of B (Q3)free fallg = 10 m s⁻²A: released from rest · B: projected horizontally · Q1: time for A to land?
The situation, not to scale. The dotted lines show that A and B are at the same height at the same moments; B just also moves sideways.

Two small balls, A and B, start together from a height of 50 m above level ground. A is released from rest (free fall). B is projected horizontally at 5 m s⁻¹. Air resistance is negligible. Take g = 10 m s⁻².

PartTask
Q1Find the time taken for ball A to hit the ground
Q2Find the horizontal distance R from the start at which ball B lands
Q3Find the displacement of ball B from its start to where it lands
Check 1 of 6

Two balls leave the same height together, one dropped from rest and one thrown sideways. Ignoring air resistance, they hit the ground…

Check 2 of 6

A ball is projected horizontally at 8 m s⁻¹ from a cliff 80 m high (g = 10 m s⁻²). The time to reach the sea is…

Check 3 of 6

For the same ball, how far from the foot of the cliff does it land?

Check 4 of 6

Ignoring air resistance, the horizontal component of a projectile’s velocity during its flight…

Check 5 of 6

Which quantity is the same for the horizontal motion and the vertical motion of a projectile?

Check 6 of 6

In Q3 the displacement of B (52.4 m) compared with its horizontal distance R = 15.8 m is…

Apply

A marble rolls off the edge of a table 1.25 m high at 2.0 m s⁻¹. Use g = 10 m s⁻². Find (a) the time to reach the floor and (b) how far from the edge of the table it lands.

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Remember earlier lessons
From lesson A.1 · 6

A body falls from rest with acceleration 10 m s⁻². After 3.0 s it has fallen…

From lesson A.1 · 7

The gradient of a velocity–time graph is…

Summary card

Key points

  • • Horizontal and vertical motion of a projectile are independent: gravity changes only the vertical motion.
  • • Horizontally: a = 0, so s = ut. Vertically: a = g, so use s = ut + ½at², v = u + at, v² = u² + 2as.
  • • The only link between the axes is the time. Find it from one axis and use it in the other.
  • • Displacement is the straight-line vector, found by Pythagoras from its two parts.

Formulas

  • s = ut + ½at²
  • R = uxt
  • d = √(R² + h²)

Key terms

Projectile
:
an object moving under gravity alone (no air resistance)
Component
:
the part of a vector along one axis

Common errors

  • • Thinking sideways speed changes the time of fall.
  • • Using the horizontal speed in the vertical equation.
  • • Giving the curved path length instead of the straight-line displacement.
Checks solved: 0 of 6
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