Kinetic Energy, Mass, and Speed
Students use data and mathematical reasoning to determine how an object's mass and speed affect its kinetic energy.
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Energy of Motion
Kinetic energy is the energy an object has because it is moving. Any moving object, from a rolling marble to a speeding bus, has kinetic energy. The amount depends on the object’s mass and speed. It is calculated with the equation KE = 1/2 mv², where KE is kinetic energy, m is mass, and v is speed. Mass is measured in kilograms, speed in meters per second, and kinetic energy in joules. For example, a 2-kilogram cart moving at 3 meters per second has KE = 1/2 × 2 × 3², or 9 joules. A stationary cart has zero kinetic energy because its speed is zero. When a force speeds up an object, work is done on it and its kinetic energy increases.
Kinetic Energy and Mass
To investigate how mass affects kinetic energy, keep speed constant and change only mass. Suppose several carts all move at 2 meters per second. A 1-kilogram cart has 2 joules of kinetic energy, a 2-kilogram cart has 4 joules, a 3-kilogram cart has 6 joules, and a 4-kilogram cart has 8 joules. These points form a straight line through the origin when kinetic energy is graphed against mass. At a constant speed, doubling the mass doubles the kinetic energy, and tripling the mass triples it. This is called a direct proportional relationship. A heavier moving object therefore has more kinetic energy than a lighter object traveling at the same speed. Keeping speed constant is important because it allows the effect of mass alone to be observed.
Kinetic Energy and Speed
To determine how speed affects kinetic energy, keep mass constant and change only speed. Consider a 2-kilogram cart. At 1 meter per second, it has 1 joule of kinetic energy. At 2 meters per second, it has 4 joules; at 3 meters per second, it has 9 joules; and at 4 meters per second, it has 16 joules. The values increase according to the square of speed because speed is squared in KE = 1/2 mv². As a result, doubling speed makes kinetic energy four times as great, not twice as great. Tripling speed makes it nine times as great. A graph of kinetic energy against speed curves upward and becomes steeper. Speed therefore has a greater effect on kinetic energy than the same numerical change in mass.
Interpreting Energy Graphs
A graph helps reveal a relationship that may be difficult to see in a data table. First, read the axis labels, units, and scale. The horizontal axis usually shows the variable being changed, such as mass or speed. The vertical axis shows kinetic energy. Next, examine the shape of the plotted data. A straight line through the origin on a kinetic-energy-versus-mass graph shows direct proportionality. An upward-curving line on a kinetic-energy-versus-speed graph shows a squared relationship. For example, on a mass graph made at 2 meters per second, a mass of 3 kilograms corresponds to 6 joules. On a speed graph for a 2-kilogram object, a speed of 3 meters per second corresponds to 9 joules. Always check which variable was held constant before comparing graphs.
Real-World Comparisons
Kinetic energy comparisons explain why mass and speed matter in transportation and safety. Imagine a bicycle and rider with a total mass of 50 kilograms moving at 4 meters per second. Their kinetic energy is 400 joules. If the same bicycle and rider increase speed to 8 meters per second, their kinetic energy becomes 1,600 joules. Doubling speed produces four times the kinetic energy. Now consider a bicycle and rider with a total mass of 100 kilograms moving at 4 meters per second. Their kinetic energy is 800 joules, twice that of the 50-kilogram system at the same speed. Greater kinetic energy means more energy must be transferred to slow down or stop. Road conditions, brakes, and reaction time also affect stopping distance, but kinetic energy helps explain why faster and more massive moving objects can be harder to stop.