A ball is thrown upward with an initial velocity of [math]80[/math] feet per second. Its distance above the ground [math]t[/math] seconds later is given by [math]s = 80t - 16t^2[/math].

• Show that it reaches its highest point when it has zero velocity.
• Show that its acceleration is constant.
• For how many seconds is it going up?
• How high does it go?
• Show that it strikes the ground with a speed of [math]80[/math] feet per second.

A particle moves on the [math]x[/math]-axis and its position [math]t[/math] seconds after it starts is given by [math]x(t) = 4t^3 - 42t^2 + 135t - 100[/math]. Describe its motion.

A particle moves on the [math]y[/math]-axis and its position [math]t[/math] seconds after it starts is given by [math]y(t) = 144t - 288 - 16t^2[/math]. Describe its motion.

A particle moves on the [math]x[/math]-axis and its position [math]t[/math] seconds after it starts is given by [math]x(t) = t^3 - 3t^2 + 5[/math]. Describe its motion.

A particle moves on the [math]y[/math]-axis and its position [math]t[/math] seconds after it starts is given by [math]y(t) = 3t^3 - 9t + 10[/math]. Describe its motion.

A particle moves on the ellipse with equation [math]4x^2 + 9y^2 = 36[/math]. When it is passing through the point [math](-3,0)[/math], what is the horizontal component of its velocity?

A particle moves on the circle with equation [math]x^2 + y^2 = 16[/math]. Show that [math]\frac{v_y}{v_x}[/math] at [math](a,b)[/math] is equal to the slope of the tangent to the circle at [math](a,b)[/math].

Two ships leave the same dock at noon, one traveling due north at [math]15[/math] knots (nautical miles per hour) and the other due east at [math]20[/math] knots. At what rate it the distance between them increasing at [math]2[/math] \textsc{p.m.}?

Water is pouring into a conical funnel and, although it is also running out of the bottom, the amount of water in the funnel is increasing at the rate of [math]3[/math] cubic inches per minute. If the conical part of the funnel is [math]5[/math] inches deep and the mouth of the funnel is [math]6[/math] inches in diameter, how fast is the water rising when it is [math]2[/math] inches deep?

• Show that, at any instant, the ratio of the rate at which the area of a circle is changing to the rate at which the radius is changing is equal to the circumference of the circle.
• Show that, at any instant, the ratio of the rate at which the volume of a sphere is changing to the rate at which the radius is changing is equal to the surface area of the sphere.