SME
Grade 11 Science · Problem type
One velocity after, the other from momentum
Use this for “A 0.6 kg cart at 0.5 m/s meets a 0.4 kg cart rolling the other way at 0.25 m/s. The first cart slows to 0.1 m/s. How fast does the other cart move, and how much kinetic energy is lost?”
Diagram
Before: m₁v₁ + m₂v₂ = 0.6 × 0.5 + 0.4 × (−0.25) = 0.2 kg·m/s
After: m₁v₁′ + m₂v₂′ = 0.6 × 0.1 + 0.4 × 0.35 = 0.2 kg·m/s
Kinetic energy: 0.0875 J before, 0.0275 J after: 0.06 J lost to heat and sound.
This is an example. Type your own values to start a new problem.
m₁ (kg)
×
v₁ (m/s)
+
m₂ (kg)
×
(
v₂ (m/s)
)
=
m₁ (kg)
×
v₁′ (m/s)
+
m₂ (kg)
×
v₂′ (m/s)
ΔKE
Kinetic energy lost
calculated · J
Formulas
m₁
×
v₁
+
m₂
×
v₂
=
m₁
×
v₁
′
+
m₂
×
v₂
′
0.6 × 0.5 + 0.4 × (−0.25) = 0.6 × 0.1 + 0.4 × 0.35
ΔKE
=
1
2
×
m₁
×
v₁
2
+
1
2
×
m₂
×
v₂
2
−
1
2
×
m₁
×
v₁
′²
−
1
2
×
m₂
×
v₂
′²
0.06
=
1
2
×
0.6
×
0.5
2
+
1
2
×
0.4
×
(−0.25)
2
−
1
2
×
0.6
×
0.1
2
−
1
2
×
0.4
×
0.35
2
Assumptions
•
No outside push along the track, so the total momentum is the same before and after.
•
+ is to the right: a cart moving left has a negative velocity.
•
Kinetic energy is not always kept: what is lost turns to heat and sound.
Step-by-step
WE KNOW
m₁ = 0.6 kg, v₁ = 0.5 m/s, m₂ = 0.4 kg, v₂ = −0.25 m/s, v₁′ = 0.1 m/s
FIND
velocity of cart 2 after (v₂′), kinetic energy lost (ΔKE)
Step 1 · Find velocity of cart 2 after (v₂′)
Use m₁ × v₁ + m₂ × v₂ = m₁ × v₁′ + m₂ × v₂′
The
momentum
before,
less
cart
1’s
after,
is
cart
2’s
after;
divide
by
m₂
.
v₂
′
=
(
m₁
×
v₁
+
m₂
×
v₂
−
m₁
×
v₁
′)/
m₂
v₂
′
=
(0.6
×
0.5
+
0.4
×
(−0.25)
−
0.6
×
0.1)/0.4
v₂
′
=
(0.3
−
0.1
−
0.06)/0.4
v₂
′
=
0.14/0.4
v₂
′
=
0.35
m/s
Step 2 · Find kinetic energy lost (ΔKE)
Use ΔKE = ½ × m₁ × v₁² + ½ × m₂ × v₂² − ½ × m₁ × v₁′² − ½ × m₂ × v₂′²
Each
cart’s
1
2
mv²
before,
less
each
cart’s
1
2
mv²
after.
ΔKE
=
1
2
×
0.6
×
0.5
2
+
1
2
×
0.4
×
(−0.25)
2
−
1
2
×
0.6
×
0.1
2
−
1
2
×
0.4
×
0.35
2
ΔKE
=
1
2
×
0.6
×
0.25
+
1
2
×
0.4
×
0.0625
−
1
2
×
0.6
×
0.01
−
1
2
×
0.4
×
0.1225
ΔKE = 0.3 × 0.25 + 0.2 × 0.0625 − 0.3 × 0.01 − 0.2 × 0.1225
ΔKE = 0.075 + 0.0125 − 0.003 − 0.0245
ΔKE = 0.06 J
Check
Put the numbers back in. Both sides should match.
0.6 × 0.5 + 0.4 × (−0.25) = 0.6 × 0.1 + 0.4 × 0.35 ✓
0.06
=
1
2
×
0.6
×
0.5
2
+
1
2
×
0.4
×
(−0.25)
2
−
1
2
×
0.6
×
0.1
2
−
1
2
×
0.4
×
0.35
2
✓
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