A soccer player takes a free kick from a spot that is $20 \mathrm{~m}$ from the goal. The ball leaves his foot at an angle of $32^{\circ}$, and it eventually hits the crossbar of the goal, which is $2.4 \mathrm{~m}$ from the ground. At what speed did the ball leave his foot?

Show that the line integral is independent of path and evaluate the integral. $\int_{C} \tan y d x+x \sec ^{2} y d y$ C is any path from $(1,0)$ to $(2, \pi / 4)$

What determines the path that an object in projectile motion follows?

In the following exercise, evaluate the integral

$$\int_C(2 x-y) d x+(x+3 y) d y$$

along the path $C$.

C : elliptic path x=4sin t, y=3cos t from (0,3) to (4,0)

True/False. If x is a unit-speed path, then $\int_{\mathbf{x}} \mathbf{F} \cdot d \mathbf{s}=\int_{a}^{b}(\mathbf{F} \cdot \mathbf{v}) d s,$ where v denotes the velocity of the path.

Tell whether each assertion is true or false. If false, give a counterexample and if true, prove it. Let G be a graph and let v and w be distinct vertices. If there is a path from v to w, there is a simple path from v to w.

What are point and path functions? Give some examples.

A person walking along a dock drags a boat by a $10-$meter rope. The boat travels along a path known as a $tractrix$ (see figure). The equation of this path is given by $y=10\ln(\frac{10+\sqrt{100-{x^{\smash{2}}}}}{x})-\sqrt{100-{x^{\smash{2}}}}$. What are the slopes of this path when $x=5$ and $x=9?$

A planet's orbit follows a path described by

$$16 x ^ { 2 } + 4 y ^ { 2 } = 64$$

A comet follows the parabolic path

$$y = x^2 - 4.$$

Where might the comet intersect the orbiting planet?

Spencer practices kicking field goals 40 yd from a goal post with a crossbar 10 ft high. If he kicks the ball with an initial velocity of 70 ft/sec at a 45° angle with the horizontal (see figure), will Spencer make the field goal if the kick sails “true”?

When baseball players throw the ball in from the outfield, they usually allow it to take one bounce before it reaches the infield, on the theory that the ball arrives sooner that way. Suppose that the angle at which a bounced ball leaves the ground is the same as the angle at which the outfielder threw it, as in Figure, but that the ball's speed after the bounce is one half of what it was before the bounce. (a) Assuming the ball is always thrown with the same initial speed, at what angle $\theta$ should the fielder throw the ball to make it go the same distance $D$ with one bounce (blue path) as a ball thrown upward at $45.0^{\circ}$ with no bounce (green path)? (b) Determine the ratio of the times for the one-bounce and no-bounce throws.

The apparent path of the sun upon the celestial sphere is called the ________.

A planet's orbit follows a path described by $16 x^2+4 y^2=64$. A comet follows the parabolic path $y=x^2-4$. Where might the comet intersect the orbiting planet?

In the following exercise, evaluate the integral

$$\int_C(2 x-y) d x+(x+3 y) d y$$

along the path $C$.

C: parabolic path $x=t, y=2 t^2$ from (0,0) to (2,8)