Unit 3 · Topic 3.1 Beta

Translational Kinetic Energy

Kinetic energy is K = ½mv², in joules.

Practice 2: Mathematical RoutinesPractice 3: Scientific Questioning and Argumentation

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Question set for this topic

Part 1 · Hook

Why this matters

A car at highway speed and the same car in a parking lot differ in one number on the speedometer, but the faster one needs far more road to stop. Kinetic energy explains why: it grows with the square of the speed, so modest speed increases have outsized consequences.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which of these is a scalar?

  1. Speed
  2. Velocity
  3. Displacement
  4. Acceleration
Show the answer

Speed has a size but no direction; the other three are vectors.

  • Correct: Speed:
  • Velocity:
  • Displacement:
  • Acceleration:

2. A cart moves west at 3.0 m/s, with east positive. What is its speed?

  1. 3.0 m/s
  2. −3.0 m/s
  3. 0 m/s
  4. 9.0 m/s
Show the answer

Speed is the size of the velocity, so it is positive: 3.0 m/s.

  • Correct: 3.0 m/s:
  • −3.0 m/s:
  • 0 m/s:
  • 9.0 m/s:

3. From v² = v₀² + 2aΔx, a car that stops (v = 0) with a constant deceleration of size a travels

  1. v₀²/(2a)
  2. v₀/(2a)
  3. 2av₀
  4. v₀²/a
Show the answer

0 = v₀² − 2aΔx, so Δx = v₀²/(2a).

  • Correct: v₀²/(2a):
  • v₀/(2a):
  • 2av₀:
  • v₀²/a:

Part 4 · See it

See it first

The same car drawn at 10 m/s and at 20 m/s on a road. At twice the speed it has four times the kinetic energy and needs four times the braking distance with the same brakes. Bars show K and 4K.
The same car at 10 m/s and at 20 m/s. Twice the speed means four times the kinetic energy and, with the same brakes, four times the stopping distance. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. An object of mass m moves at speed vit has kinetic energy K = ½mv², measured in joules (kg·m²/s²)
  2. The speed enters squareddoubling the speed makes K four times as large, while doubling the mass only doubles it
  3. Squaring removes the sign of the velocityK is a scalar that is never negative and does not depend on direction
  4. Speed depends on the reference framethe same object has different kinetic energies in different frames
  5. A system has several moving partsits kinetic energy is the sum of the parts' kinetic energies, which never cancel

Part 6 · Key ideas

Key ideas

  • Energy is a scalar quantity a system has; it can change form and move between objects. Its SI unit is the joule: 1 J = 1 kg·m²/s².
  • Kinetic energy, K = ½mv², is the energy of motion. It depends on mass and on the speed squared.
  • K is never negative and has no direction. A cart moving left at 3 m/s has the same K as one moving right at 3 m/s.
  • ΔK = ½mv_f² − ½mv_i², which is not ½m(Δv)². Square each speed first, then subtract.
  • Kinetic energy depends on the reference frame, because speed does.

Part 7 · Misconception

A common mistake

The wrong idea: A cart that speeds up from 2.0 m/s to 3.0 m/s gains ½m(1.0 m/s)² of kinetic energy.

What actually happens: The change is ½m(3.0 m/s)² − ½m(2.0 m/s)² = ½m(5.0 m²/s²), five times ½m(Δv)² = ½m(1.0 m²/s²). Square each speed before you subtract.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

One car at two speeds

The same car at two speeds, braking on the same roadv = 10 m/sv = 20 m/s

The car has a mass of 1200 kg. In both rows the driver brakes with the same braking force.

1. What is the car's kinetic energy at 10 m/s? Give the value and its unit.

Type a number and its unit.

Show the answer

K = ½mv² = ½(1200 kg)(10 m/s)² = 60,000 J = 6.0 × 10⁴ J. A joule is a kg·m²/s².

  • Answer: 6.0 × 104 J

2. How does the car's kinetic energy at 20 m/s compare with its kinetic energy at 10 m/s?

  1. 4 times as large
  2. 2 times as large
  3. 8 times as large
  4. The same, since the mass is the same
Show the answer

K = ½mv². The mass is fixed, so K ∝ v². Doubling v multiplies K by 2² = 4.

  • Correct: 4 times as large: Right: K depends on v², and (20/10)² = 4.
  • 2 times as large: This treats K as proportional to v. The speed is squared, so doubling it multiplies K by 4.
  • 8 times as large: This cubes the speed ratio. K = ½mv² has v squared, not cubed.
  • The same, since the mass is the same: The mass is the same, but K also depends on the speed, and the speed doubled.

3. By how much does the car's kinetic energy increase when it speeds up from 10 m/s to 20 m/s? Give the value and its unit.

Type a number and its unit.

Show the answer

ΔK = ½mv_f² − ½mv_i² = ½(1200 kg)[(20 m/s)² − (10 m/s)²] = ½(1200)(300) J = 1.8 × 10⁵ J. Note this is three times ½m(Δv)².

  • Answer: 1.8 × 105 J

4. With the same braking force, the car at 10 m/s stops in a distance d. In what distance does it stop from 20 m/s?

  1. 2d
  2. 4d
  3. d
  4. √2 d
Show the answer

The same braking force gives the same deceleration a (Newton's second law). From v² = v₀² + 2aΔx with v = 0, Δx = v₀²/(2a), so doubling v₀ makes the distance 4 times as long, matching the four times larger kinetic energy.

  • 2d: This assumes the stopping distance grows in step with the speed. With constant deceleration it grows with v².
  • Correct: 4d: Right: the same force gives the same deceleration, and v² = 2aΔx makes the distance grow as v².
  • d: The deceleration is the same, but a faster car needs more distance to lose its speed.
  • √2 d: This takes a square root of the speed ratio. The distance depends on v², so the factor is 2² = 4.

5. A student finds that a 2.0 kg cart moving left at 3.0 m/s has a kinetic energy of −9.0 J. What went wrong?

  1. Nothing: leftward motion has negative kinetic energy
  2. The ½ was left out; it should be −4.5 J
  3. K cannot be negative; it is +9.0 J
  4. The speed should not be squared; it is −3.0 J
Show the answer

K = ½mv² = ½(2.0 kg)(−3.0 m/s)² = +9.0 J. Kinetic energy is a scalar and is never negative.

  • Nothing: leftward motion has negative kinetic energy: Kinetic energy has no direction. Squaring the velocity removes its sign.
  • The ½ was left out; it should be −4.5 J: The ½ was used: ½(2.0)(9.0) = 9.0. The real slip is the sign.
  • Correct: K cannot be negative; it is +9.0 J: Right: (−3.0 m/s)² is positive, so K = +9.0 J.
  • The speed should not be squared; it is −3.0 J: K = ½mv² does square the speed. The answer is positive.

6. A crate moving at 3.0 m/s has 54 J of kinetic energy. What is its mass, in kg?

Type a number in kg.

Show the answer

m = 2K/v² = 2(54 J)/(3.0 m/s)² = 108/9.0 kg = 12 kg.

  • Answer: 12 kg

Part 9 · Summary

Summary

Kinetic energy is K = ½mv², in joules. It is a scalar, never negative, independent of direction and dependent on the reference frame. Because the speed is squared, doubling the speed quadruples K; and a change in K is ½mv_f² − ½mv_i², not ½m(Δv)².

Part 10 · Up next

What comes next

Part 11 · Connections

Connections