The height of a projectile fired in the air vertically with...

The height of a projectile fired in the air vertically with initial velocity 25 $\mathrm{m} / \mathrm{s}$ is $$h(t)=25 t-4.9 t^{2} \mathrm{m}.$$ (a) Compute $h(1) .$ Show that $h(t)-h(1)$ can be factored with $(t-1)$ as a factor. (b) Using part (a), show that the average velocity over the interval $[1, t]$ is $20.1-4.9 t.$ (c) Use this formula to find the average velocity over several intervals $[1, t]$ with $t$ close to $1 .$ Then estimate the instantaneous velocity at time $t=1 .$

The height of a projectile fired in the air vertically with initial velocity 25 $\mathrm{m} / \mathrm{s}$ is
$$h(t)=25 t-4.9 t^{2} \mathrm{m}.$$
(a) Compute $h(1) .$ Show that $h(t)-h(1)$ can be factored with $(t-1)$ as a factor.
(b) Using part (a), show that the average velocity over the interval $[1, t]$ is $20.1-4.9 t.$
(c) Use this formula to find the average velocity over several intervals $[1, t]$ with $t$ close to $1 .$ Then estimate the instantaneous velocity at time $t=1 .$

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of its factors. This technique is useful for simplifying expressions, solving equations, or revealing hidden differences of values that allow for canceling common factors, such as when demonstrating that an expression has a particular linear factor.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. It is calculated as the limit of the average velocity as the time interval shrinks to zero, which is equivalent to the derivative of the position function with respect to time.

Difference Quotient

The difference quotient is a formula used to compute the average rate of change of a function over an interval. It is expressed as the ratio of the difference in function values to the difference in the independent variable values. In the context of the derivative, taking the limit of the difference quotient as the interval approaches zero yields the instantaneous rate of change.

Projectile Motion

Projectile motion refers to the movement of an object that is launched into the air and influenced primarily by gravity. It is typically characterized by quadratic equations that describe the height or the horizontal distance traveled as functions of time, incorporating the effects of initial velocity and gravitational acceleration.

Average Velocity

Average velocity is defined as the change in position (or height) divided by the elapsed time over a given time interval. In calculus, it serves as the slope of the secant line between two points on a function, providing an approximate rate of change of the object's position over that interval.

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