2.5. The Kinematic Equations http://www.ck12.org
Check Your Understanding
- A bus accelerates at 1.5 m/s^2 from rest, due east. What is the instantaneous speed of the bus at 4.0 s? What is the
instantaneous velocity of the bus at 4.0 s?
Answer:We begin by realizing that “from rest” means the initial velocity (and speed) of the bus is zero. Concep-
tually, we imagine that every second the bus gains 1.5 m/s of additional speed. Therefore, at the end of the fourth
second, the bus has a speed of 6.0 m/s and a velocity of 6.0 m/s due east.
- A toy rocket is launched straight upward with an initial velocity of + 30.0 m/s. If the time interval from launch to
highest position reached is 3.0 s, what is the magnitude and direction of the rocket’s acceleration during its upward
flight?
Answer:At its highest position above ground, before falling back to Earth, its velocity is 0.0 m/s.
a=∆∆tv=((vtff−−vtii)),a=(^0 −( 330 ))=−10 m/s^2
- A toy rocket falls from the top point, straight down, landing with a velocity of -30.0 m/s. If the time interval from
release to ground level is 3.0 s, what is the magnitude and direction of the rocket’s acceleration during its downward
flight?
Answer:a=∆∆vt=((vtff−−tvii)),a=(−^30 ( 3 −)^0 )=−10 m/s^2
Questions to consider:
- What causes the rocket’s acceleration?
- Why is the acceleration the same whether the rocket is ascending or descending?
Gravitational Acceleration
The rate at which a freely falling object increases its velocity is attributed to the gravitational force that the Earth
exerts on bodies near its surface. The force of gravity near the Earth’s surface is assumed to be constant, and
therefore the gravitational acceleration near the Earth’s surface is constant, as well. It was Galileo who challenged
the accepted wisdom of Aristotle. Aristotle hypothesized that objects fell at a rate proportional to their weight.
Galileo, on the other hand, insisted that all objects, regardless of their weight, in the absence of air resistance, fall at
the same rate. See the link below regarding Galileo’s famous ball drop experiment.
http://demonstrations.wolfram.com/GalileosExperimentAtTheLeaningTowerOfPisa/
Galileo Galilei (1564-1642) was the first person to determine the numerical value of the acceleration of gravity near
the Earth’s surface. In order to do this, he needed to convince himself that when an object was released, it continually
sped up. Early on, he believed that after a very short time, the speed of a falling object remained fixed. It was not until
he “diluted” the effect of gravity by rolling balls down long inclined planes in order to decrease the acceleration of
gravity that he convinced himself that objects not only continue to increase their speed, but that they increased their
speed by a fixed amount during equal time intervals. For example, a solid ball rolling down an inclined plane fixed
at a five degree angle increases its speed about 0.61 m/s every second. That is, it has an acceleration of 0.61 m/s^2.
Galileo was able to reason that when the inclined plane is raised to 90 degrees (that is, vertical) falling objects would
accelerate at−10 m/s^2. This is the value of the gravitational acceleration,g, near the Earth’s surface.
The exact value of gravitational acceleration varies with your position on the Earth, going from− 9 .75 m/s^2 to− 9 .83 m/s^2 ,
but the value of -10 will be used for our calculations.
The acceleration-time graph below is for a toy rocket launched upward with an initial velocity of +30 m/s. Note that
the initial velocitycannotbe determined from the graph—it must be a supplied piece of information; or, as we say,
“an initial condition.”