Velocity and Position Can Be Derived from Acceleration
The braking distance is the distance traveled from the moment the brakes are applied until the car stops. In this model, it is 8.8 m at 40 km/h.
Adjust conditions:
The sliders adjust the values
2台の減速度が同じで一定なら、80 km/hの制動距離はいくらか？
2台とも0.75秒後に制動。80 km/hの停止距離は40 km/hの何倍か？
The equations explain the phenomenon
1Average velocity is definedvaverage=ΔxΔt……
2Instantaneous velocity is obtained by shortening the time interval Δtv=dxdt……
3Acceleration is defined from the change in velocitya=dvdt……
4The mass and net force are constantdvdt=Fm=a (constant)……
The following example uses the 40 km/h car above (11.1 m/s), which stops with a constant deceleration of 7.0 m/s².
5Velocity is obtained by integrating accelerationv=v₀+a t……
6Position is obtained by integrating velocityx=x₀+v₀ t+½ a t²……
7The sum includes intervals with negative velocityx − x₀=(Area above the time axis)−(Area below the time axis)……
8Eliminating time t relates velocity to positionv² − v₀²=2 a (x − x₀)……
Further applications cover free fall, upward throws, and measurement
9The equations are applied to free fally=½ g t²……
10An object is thrown vertically upwardy=v₀ t−½ g t²……
11Investigating how errors in time measurement affect the calculated value of gg=2yt²……
What this chapter has shown
Over a short time interval Δt, the velocity can be treated as approximately constant, so the change in position is approximated by vΔt. This is the signed area of a rectangle on the v−t graph. Dividing the interval from time 0 to t into increasingly small intervals and adding their contributions gives x−x₀=∫₀ᵗv(τ)dτ. The area below the time axis contributes negatively.
4.00 timesBraking distance ratio when the speed doubles
For the same constant deceleration, doubling the speed increases the braking distance by a factor of 4. If both cars begin braking after 0.75 seconds, the total stopping distance increases by a factor of 3.03.
In everyday lifeThe length of skid marks provides a clue to a car's speed. Estimating the speed requires information about conditions such as road friction; the length of the marks alone does not determine the speed.
Velocity is the derivative of position, and acceleration is the derivative of velocity. Conversely, integrating acceleration gives velocity, and integrating velocity gives position. When the force is constant, acceleration is also constant, so the 3 equations of motion for constant acceleration follow simply from integrating the equation of motion 2 times. The slope of a v−t graph gives acceleration, and the area under the graph gives the change in position. These interpretations show that all 3 equations describe the same motion.