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Grade 10-11 (age 15-17)

high school physics Uniformly Accelerated Motion

Like a car pulling away when the light changes, motion whose velocity changes by the same amount every second is called uniformly accelerated motion. With a constant acceleration the v-t graph is a sloping straight line; its slope is the acceleration, and the area between the graph and the time axis is the displacement. Of the three formulas linking velocity, displacement and time, pick the one that leaves out the quantity you do not need, and common problems such as free fall work out at once. Move the initial velocity and acceleration sliders and the slope and area of the v-t line change; make the acceleration negative and the stretch where the velocity passes 0 and the object moves backward appears as a red area. The free-fall figure uses g fixed at 10 m/s².

Motion with Steadily Changing Velocity

🏎️ The idea of uniform acceleration
①A car pulling away when the light changes keeps speeding up
②With a constant acceleration, the velocity changes by the same amount every second
③With a = 3 m/s² the velocity grows by 3 m/s each second, and with a negative a it shrinks by that much

Uniform Acceleration on a v-t Graph

2 m/s
3 m/s²
📊 Reading the graph
①A straight v-t graph means uniformly accelerated motion
②The slope is the acceleration a; as the dashed triangle shows, the velocity changes by a in 1 s
③The area between the graph and the time axis is the displacement s
④The area after the velocity turns negative (red) is distance covered backward, so it is subtracted from the displacement

The Three Formulas

Velocity
v = v₀ + at
Final velocity = initial velocity + acceleration × time
Displacement
s = v₀t + 12at²
Displacement = initial velocity × time + ½ × acceleration × time²
Velocity and displacement
v² = v₀² + 2as
The relation between velocity and displacement without the time t
💡 How to choose a formula
①If the displacement s is not needed, use v = v₀ + at
②If the final velocity v is not needed, use s = v₀t + ½at²
③If the time t is not needed, use v² = v₀² + 2as

Free Fall

Free fall
v = gt, s = 12gt² (g ≈ 9.8 m/s²)
Uniformly accelerated motion with initial velocity v₀ = 0 and acceleration g. The figure fixes g = 10 m/s²; problems use 9.8 or 10 as stated

Worked Examples

Example 1
An object with initial speed 5 m/s accelerates uniformly at 2 m/s². What is its speed after 4 s?
1
The final velocity is found from the time and acceleration, so use v = v₀ + at.
v = v₀ + at
2
Put in v₀ = 5, a = 2 and t = 4.
v = 5 + 2 × 4 = 13 m/s
▸ 13 m/s
No displacement is needed, so the simplest velocity formula is enough.
Example 2
An object is released from rest and falls freely for 3 s. How far does it fall? Use g = 10 m/s².
1
Free fall is uniformly accelerated motion with v₀ = 0, so use s = ½gt².
s = 12gt2
2
Put in g = 10 and t = 3.
s = 12 × 10 × 32 = 45 m
▸ 45 m
The distance fallen is proportional to the square of the time: 45 m in 3 s is 9 times the 5 m in 1 s.

Summary

Velocity
v = v₀ + at
Displacement
s = v₀t + 12at²
Velocity and displacement
v² = v₀² + 2as
The formula with time removed
CSAT-style problem, adapted
A car moving at 20 m/s on a straight road brakes and stops after travelling 40 m more. What is its acceleration?
①−2 m/s²
②−4 m/s²
③−5 m/s²
④−8 m/s²
⑤−10 m/s²
▸ ③ −5 m/s²
1
The time t is unknown, so use v² = v₀² + 2as, with v = 0 because the car stops.
0 = v₀2 + 2as
2
Put in v₀ = 20 and s = 40 and solve for a.
400 + 80a = 0 ⇒ a = −5 m/s2
🎯 Exam Points
①Choose among v = v₀ + at, s = v₀t + ½at² and v² = v₀² + 2as to fit the situation
②On a v-t graph the slope is the acceleration and the area is the displacement
③Free fall has v₀ = 0 and a = g, and the distance fallen in each second grows as 1 : 3 : 5 : 7
④An object thrown upward has v₀ > 0 and a = −g, so gravity slows it down
⑤The average velocity (v₀ + v)/2 can be used only for uniform acceleration
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Uniform Linear Motion
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Newton's Laws of Motion

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