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# 15.2 Acceleration

15.16   Definition (Acceleration.) If a particle moves in a straight line so that its position at time is , we have defined its velocity at time to be . We now define its acceleration at time to be , so that acceleration is the derivative of velocity. Thus if a particle moves with a constant acceleration of , then every second its velocity increases by one ft./sec.

15.17   Example. A mass on the end of a spring moves so that its height at time is , where and are positive numbers. If we denote its velocity at time by , and its acceleration at time by then

From this we see that the acceleration is always of opposite sign from the position: when the mass is above the zero position it is being accelerated downward, and when it is below its equilibrium position it is being accelerated upward. Also we see that the magnitude of the acceleration is largest when the velocity is .

15.18   Definition (Acceleration due to gravity.) If a particle moves near the surface of the earth, acted on by no forces except the force due to gravity, then will move with a constant acceleration which is independent of the mass of . The value of is

We call the acceleration due to gravity. Actually, the value of varies slightly over the surface of the earth, so there is no exact value for . The law just described applies in situations when air resistance and buoyancy can be neglected. It describes the motion of a falling rock well, but it does not describe a falling balloon.

Remark: When I solve applied problems, I will usually omit all units (e.g. feet or seconds) in my calculations, and will put them in only in the final answers.

15.19   Example. A juggler tosses a ball vertically upward from a height of feet above the ground with a speed of ft./sec. Let denote the height of the ball above the ground at time . We will set our clock so that corresponds to the time of the toss:

We will suppose that while the ball is in the air, its motion is described by a differentiable function of . We assume that

We know one function whose derivative is :

By the antiderivative theorem it follows that there is a constant such that

Moreover we can calculate as follows:

Thus

We know a function whose derivative is :

Thus there is a constant such that

To find we set :

Thus

The ball will reach its maximum height when , i.e. when

The maximum height reached by the ball is

so the ball rises to a maximum height of feet above the ground.

15.20   Example (Conservation of energy.) Suppose that a particle moves near the surface of the earth acted upon by no forces except the force of gravity. Let and denote respectively its height above the earth and its velocity at time . Then

so

Now

so we have

It follows that there is a constant such that

or

If is the mass of the particle then
 (15.21)

The quantity is called the kinetic energy of , and the quantity is called the potential energy of . Equation (15.21) states that as moves, the sum of its potential energy end its kinetic energy remains constant.

15.22   Exercise. A A particle moves in a vertical line near the surface of the earth, acted upon by no forces except the force of gravity. At time it is at height , and has velocity . Derive the formula for the height of the particle at time .

15.23   Exercise. The acceleration due to gravity on the moon is approximately

where denotes the acceleration due to gravity on the earth. A juggler on the moon wants to toss a ball vertically upward so that it rises 4 feet above its starting height. With what velocity should the ball leave 's hand?

Next: 15.3 Convexity Up: 15. The Second Derivative Previous: 15.1 Higher Order Derivatives   Index
Ray Mayer 2007-09-07