A shot leaves the throwers hand 6.5 ft above the ground at

In Exercises 1 and 2, graph the curves and sketch their velocity and acceleration vectors at the given values of t. Then write a in the form without finding T and N, and find the value of at the given values of t

In Exercises 1 and 2, graph the curves and sketch their velocity and acceleration vectors at the given values of t. Then write a in the form without finding T and N, and find the value of at the given values of t

The position of a particle in the plane at time t is Find the particles highest speed

. Suppose Show that the angle between r and a never changes. What is the angle?

At point P, the velocity and acceleration of a particle moving in the plane are and Find the curvature of the particles path at P

Find the point on the curve where the curvature is greatest.

A particle moves around the unit circle in the xy-plane. Its position at time t is where x and y are differentiable functions of t. Find dy dt if Is the motion clockwise or counterclockwise?

You send a message through a pneumatic tube that follows the curve (distance in meters). At the point (3, 3), and Find the values of and at (3, 3)

A particle moves in the plane so that its velocity and position vectors are always orthogonal. Show that the particle moves in a circle centered at the origin

A circular wheel with radius 1 ft and center C rolls to the right along the x-axis at a half-turn per second. (See the accompanying figure.) At time t seconds, the position vector of the point P on the wheels circumference is a. Sketch the curve traced by P during the interval 0 t 3. b. Find v and a at and 3 and add these vectors to your sketch. c. At any given time, what is the forward speed of the topmost point of the wheel? Of C?

A shot leaves the throwers hand 6.5 ft above the ground at a 45 angle at 44 ft sec. Where is it 3 sec later?

A javelin leaves the throwers hand 7 ft above the ground at a 45 angle at 80 ft sec. How high does it go?

A golf ball is hit with an initial speed at an angle to the horizontal from a point that lies at the foot of a straight-sided hill that is inclined at an angle to the horizontal, where Show that the ball lands at a distance measured up the face of the hill. Hence, show that the greatest range that can be achieved for a given occurs when i.e., when the initial velocity vector bisects the angle between the vertical and the hill

In Potsdam in 1988, Petra Felke of (then) East Germany set a womens world record by throwing a javelin 262 ft 5 in. a. Assuming that Felke launched the javelin at a 40 angle to the horizontal 6.5 ft above the ground, what was the javelins initial speed? b. How high did the javelin go?

Find the lengths of the curves in Exercises 15 and 16.

Find the lengths of the curves in Exercises 15 and 16.

In Exercises 1720, find T, N, B, and at the given value of t.

In Exercises 1720, find T, N, B, and at the given value of t.

In Exercises 1720, find T, N, B, and at the given value of t.

In Exercises 1720, find T, N, B, and at the given value of t.

In Exercises 21 and 22, write a in the form at without finding T and N.

In Exercises 21 and 22, write a in the form at without finding T and N.

Find T, N, B, and as functions of t if

At what times in the interval are the velocity and acceleration vectors of the motion orthogonal?

The position of a particle moving in space at time is Find the first time r is orthogonal to the vector

Find equations for the osculating, normal, and rectifying planes of the curve at the point (1, 1, 1).

Find parametric equations for the line that is tangent to the curve at

Find parametric equations for the line tangent to the helix at the point where

By eliminating from the ideal projectile equations show that This shows that projectiles launched simultaneously from the origin at the same initial speed will, at any given instant, all lie on the circle of radius centered at regardless of their launch angle. These circles are the synchronous curves of the launching.

Show that the radius of curvature of a twice-differentiable plane curve is given by the formula

An alternative definition gives the curvature of a sufficiently differentiable plane curve to be where is the angle between T and i (Figure 13.37a). Figure 13.37b shows the distance s measured counterclockwise around the circle from the point (a, 0) to a point P, along with the angle at P. Calculate the circles curvature using the alternative definition. (Hint: f = u + p>2.)

What percentage of Earths surface area could the astronauts see when Skylab 4 was at its apogee height, 437 km above the surface? To find out, model the visible surface as the surface generated by revolving the circular arc GT, shown here, about the y-axis. Then carry out these steps: 1. Use similar triangles in the figure to show that Solve for 2. To four significant digits, calculate the visible area as 3. Express the result as a percentage of Earths surface area