Example 1 — Projectile motion

A ball is launched from ground level at 20 m s⁻¹ at 30° above the horizontal. Take g = 9.81 m s⁻². Find the time of flight and the horizontal range.

Split the initial velocity into components. The horizontal and vertical motions are independent and share only the time.

uₓ = 20 cos 30° = 17.3 m s⁻¹      (constant throughout)
u_y = 20 sin 30° = 10.0 m s⁻¹     (decelerating at g)

Time of flight. The ball returns to its launch height, so vertical displacement s = 0. Using s = ut + ½at² with a = −9.81:

0 = 10.0t − 4.905t²
0 = t(10.0 − 4.905t)
t = 0  or  t = 10.0 / 4.905 = 2.04 s

t = 0 is the launch. The flight time is 2.04 s.

Range. Horizontally there is no acceleration, so distance is simply speed × time:

R = uₓ t = 17.3 × 2.04 = 35.3 m

Time of flight 2.04 s; range 35.3 m.

Note the shortcut and why it is safe here: at the same launch height, time up equals time down, so t = 2u_y/g = 20.0/9.81 = 2.04 s gives the same result in one line. It is only valid when the landing height equals the launch height — off a cliff, you must solve the quadratic.

Example 2 — An inelastic collision

A trolley of mass 2.0 kg moving at 3.0 m s⁻¹ collides with a stationary 3.0 kg trolley and the two move off together. (a) Find their common velocity. (b) Determine whether the collision is elastic.

(a) Momentum is conserved in every collision. Take motion to the right as positive:

p before = (2.0 × 3.0) + (3.0 × 0) = 6.0 kg m s⁻¹
p after  = (2.0 + 3.0) v = 5.0v

5.0v = 6.0
v = 1.2 m s⁻¹  to the right

(b) Compare kinetic energy before and after — this is the only test for elasticity:

Ek before = ½ × 2.0 × 3.0² = 9.0 J
Ek after  = ½ × 5.0 × 1.2² = 3.6 J

3.6 J < 9.0 J, so 5.4 J of kinetic energy has been lost. The collision is inelastic.

The energy is not destroyed — it has become internal energy of the trolleys, plus sound. Saying "energy was lost" without saying where it went often costs a mark, because it invites the reading that conservation of energy was violated.

Do not test elasticity by checking momentum. Momentum is conserved either way, so it tells you nothing about which type of collision this is. Students confuse this constantly.

Example 3 — Forces on an incline

A 5.0 kg block rests on a frictionless plane inclined at 25° to the horizontal. Find the component of its weight acting down the slope, and its acceleration.

Resolve the weight along and perpendicular to the slope:

W = mg = 5.0 × 9.81 = 49.05 N

Component along slope       = mg sin θ = 49.05 × sin 25° = 20.7 N
Component perpendicular     = mg cos θ = 49.05 × cos 25° = 44.5 N

The perpendicular component is balanced by the normal force, so the resultant is the 20.7 N along the slope:

a = F/m = 20.7 / 5.0 = 4.15 m s⁻²   down the slope

Equivalently a = g sin θ = 9.81 × sin 25° = 4.15 m s⁻², independent of mass.

20.7 N; 4.15 m s⁻² down the slope.

How to be certain about sine and cosine. Imagine the slope flattening to θ = 0. The component down the slope must vanish, and sin 0° = 0 — so the slope component takes the sine. Two seconds of checking removes the most common error in this section permanently.

Example 4 — Work, energy and power

A car of mass 1200 kg accelerates from rest to 25 m s⁻¹ in 8.0 s. Calculate the gain in kinetic energy and the average power developed.

ΔEk = ½mv² − ½mu²
    = ½ × 1200 × 25² − 0
    = 600 × 625
    = 3.75 × 10⁵ J

P = W/t = 3.75 × 10⁵ / 8.0 = 4.69 × 10⁴ W  (46.9 kW)

3.75 × 10⁵ J; 4.69 × 10⁴ W.

This is the average power, and the question said so. The instantaneous power at the end, from P = Fv, is considerably higher — the car needs more power at 25 m s⁻¹ than at 5 m s⁻¹ for the same force. If a question asks for instantaneous power, P = Fv is the route; if it asks for average, go through the energy.

Note also that this is the power delivered to the car's kinetic energy alone. The engine's output is larger, because some is lost to drag and friction — which is what an efficiency question would be getting at.

Educerie · original worked examples written against the published IB syllabus structure for Physics Theme A, first assessment 2025. All scenarios and values are our own. Last reviewed 5 September 2026.