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Although most physicists would agree that the concept of force is not as fundamental as the concept of a conservation law, it is still considered central to the study of physics. A force is a push, a pull, or any other interaction, exerted by one object on another object. We know from experience that a push or a pull often causes an object to move. This allows us to quantify the definition of force in terms of a quantity that was defined previously: acceleration. Restart.

If an object's mass remains constant, the magnitude of a force exerted on an object is proportional to the time rate of change of the velocity (i.e., acceleration). Specifically, Σ F = m a.

Use this definition as you consider the results of Illustration 4.3 (position is given in meters and time is given in seconds). Set the mass in the text box before you select the graph type, velocity or acceleration.

The two-handed image ("handy") interacts with the 1.0-kg cart in the animation if the image is near the left-hand or right-hand end of the cart. The arrow below the cart shows the direction and strength of the force exerted on the cart. You will have to move the image to keep it behind the cart since the interaction changes direction if the image passes through the center of the cart. Start the animation and explore it for a few minutes. Reset the animation if the cart goes off the end of the track.

Now select velocity (and then acceleration). Drag the handy image to the left of the cart and try to apply the force for as brief a period of time as you can. This will result in a force applied to the cart only for a short period of time and then no force will act. What do the resulting velocity and acceleration graphs look like? The velocity graph should show an increasing velocity for the instant handy is acting on the cart; then it should have a slope of zero. The velocity only changes when the force is acting. The acceleration graph should give a spike during the application of the force and be zero otherwise. Repeat the same process when the image is to the right of the cart. What changes? Because force is a vector, the applied force is now in the negative x direction. Therefore, the velocity and the acceleration are now both negative as well.

Now select velocity (and then acceleration). Drag the handy image to the left of the cart and then keep dragging it to the right as the cart moves. This will result in a constant force applied to the cart. What do the resulting velocity and acceleration graphs look like? The velocity graph should have a constant slope upward while the acceleration graph should give a constant acceleration during the application of the force. Repeat the process when the image is to the right of the cart. What changes? Because force is a vector, the applied force is now in the negative x direction. Therefore, the velocity and the acceleration are now both negative as well.

What changes on the velocity and acceleration graphs will occur if the mass of the cart is doubled or decreased by a factor of two? Try it and find out. Since acceleration is equal to the force over the mass, an increase in mass means a smaller acceleration, and a decrease in mass means a larger acceleration.