Use the table below to calculate the difference quotients for h = −0.2, −0.1, 0.1, 0.2. Then estimate the velocity and speed at t = 1.

Draw the vectors r(2 + h) − r(2) and for h = 0.5 for the path in Figure 10. Draw v(2) (using a rough estimate for its length).

Find a(t) for a particle moving around a circle of radius 8 cm at a constant speed of υ = 4 cm/s (see Example 4). Draw the path and acceleration vector at .

Sketch the path r(t) = 〈1 − t², 1 − t〉 for −2 ≤ t ≤ 2, indicating the direction of motion. Draw the velocity and acceleration vectors at t = 0 and t = 1.

Sketch the path r(t) = 〈t², t³〉 together with the velocity and acceleration vectors at t = 1.

The paths r(t) = 〈t², t³〉 and r₁(t) = 〈t⁴, t⁶〉 trace the same curve, and r₁(1) = r(1). Do you expect either the velocity vectors or the acceleration vectors of these paths at t = 1 to point in the same direction? Compute these vectors and draw them on a single plot of the curve.

A bullet is fired from the ground at an angle of 45°. What initial speed must the bullet have in order to hit the top of a 120-m tower located 180 m away?

Find the initial velocity vector v₀ of a projectile released with initial speed 100 m/s that reaches a maximum height of 300 m.

Show that a projectile fired at an angle θ with initial speed υ₀ travels a total distance sin 2θ before hitting the ground. Conclude that the maximum distance (for a given υ₀) is attained for θ = 45°.

One player throws a baseball to another player standing 25 m away with initial speed 18 m/s. Use the result of Exercise 21 to find two angles θ at which the ball can be released. Which angle gets the ball there faster?

A bullet is fired at an angle at a tower located d = 600 m away, with initial speed υ₀ = 120 m/s. Find the height H at which the bullet hits the tower.

Show that a bullet fired at an angle θ will hit the top of an h-meter tower located d meters away if its initial speed is

A constant force F = 〈5, 2〉 (in newtons) acts on a 10-kg mass. Find the position of the mass at t = 10 s if it is located at the origin at t = 0 and has initial velocity v₀ = 〈2, −3〉 (in meters per second).

A force F = 〈24t, 16 − 8t〉 (in newtons) acts on a 4-kg mass. Find the position of the mass at t = 3 s if it is located at (10, 12) at t = 0 and has zero initial velocity.

A particle follows a path r(t) for 0 ≤ t ≤ T, beginning at the origin O. The vector is called the average velocity vector. Suppose that . Answer and explain the following:

At a certain moment, a moving particle has velocity v = 〈2, 2, −1〉 and a = 〈0, 4, 3〉. Find T, N, and the decomposition of a into tangential and normal components.

At a certain moment, a particle moving along a path has velocity v = 〈12, 20, 20〉 and acceleration a = 〈2, 1, −3〉. Is the particle speeding up or slowing down?

r(t) = 〈t², t³〉

r(t) = 〈t, cos t, sin t〉

Let r(t) = 〈t², 4t − 3〉. Find T(t) and N(t), and show that the decomposition of a(t) into tangential and normal components is

Find the components a_T and a_N of the acceleration vector of a particle moving along a circular path of radius R = 100 cm with constant velocity υ₀ = 5 cm/s.

In the notation of Example 5, find the acceleration vector for a person seated in a car at (a) the highest point of the Ferris wheel and(b) the two points level with the center of the wheel.

Suppose that the Ferris wheel in Example 5 is rotating clockwise and that the point P at angle 45° has acceleration vector a = 〈0, −50〉 m/min² pointing down, as in Figure 11. Determine the speed and tangential acceleration of the Ferris wheel.

At time t₀, a moving particle has velocity vector v = 2i and acceleration vector a = 3i + 18k. Determine the curvature κ(t₀) of the particle’s path at time t₀.

A space shuttle orbits the earth at an altitude 400 km above the earth’s surface, with constant speed υ = 28,000 km/h. Find the magnitude of the shuttle’s acceleration (in km/h²), assuming that the radius of the earth is 6378 km (Figure 12).

A car proceeds along a circular path of radius R = 300 m centered at the origin. Starting at rest, its speed increases at a rate of t m/s². Find the acceleration vector a at time t = 3 s and determine its decomposition into normal and tangential components.

A runner runs along the helix r(t) = 〈cos t, sin t, t〉. When he is at position , his speed is 3 m/s and he is accelerating at a rate of . Find his acceleration vector a at this moment. Note: The runner’s acceleration vector does not coincide with the acceleration vector of r(t).

Explain why the vector w in Figure 13 cannot be the acceleration vector of a particle moving along the circle. Hint: Consider the sign of w · N.

Figure 14 shows acceleration vectors of a particle moving clockwise around a circle. In each case, state whether the particle is speeding up, slowing down, or momentarily at constant speed. Explain.

Prove that .

Suppose that r = r(t) lies on a sphere of radius R for all t. Let J = r × r′. Show that r′ = (J × r)/∥r∥². Hint: Observe that r and r′ are perpendicular.