Chapter 4: Kinematics in Two Dimensions

A spaceship maneuvering near Planet Zeta is located at$\widehat{r}=(600\widehat{i}-400\widehat{j}+200\widehat{k})\times {10}^3\text{km}$ , relative to the planet, and traveling at $\overrightarrow{v}=9500\widehat{ı}\text{m}/\text{s}$. It turns on its thruster engine and accelerates with $\overrightarrow{a}=(40\widehat{ı}-20\widehat{k})\text{m}/{\text{s}}^2$for $35\text{min}$. What is the spaceship's position when the engine shuts off? Give your answer as a position vector measured in $\text{km}$.

A particle moving in the xy--plane has velocity ${\overrightarrow{v}}_0=v_{0x}\widehat{ı}+v_{0y}\widehat{J}$CALC at$t=0$ . It undergoes acceleration , where $b$and $c$are constants. Find an expression for the particle's velocity at a later time $t$.

A projectile’s horizontal range over level ground is $\raisebox{1ex}{${v_0}^2\sin 2\theta $}\!\left/ \!\raisebox{-1ex}{$g$}\right.$. At what launch angle or angles will the projectile land at half of its maximum possible range?

a. A projectile is launched with speed $v_0$and angle $\theta$. Derive an

expression for the projectile’s maximum height $h$.

b. A baseball is hit with a speed of 33.6 m/s. Calculate its

height and the distance traveled if it is hit at angles of

30.0°, 45.0°, and 60.0°.

A projectile is launched from ground level at angle $\theta$and speed

$v_0$into a headwind that causes a constant horizontal acceleration

of magnitude $a$opposite the direction of motion.

a. Find an expression in terms of $a$and $g$for the launch angle

that gives maximum range.

b. What is the angle for maximum range if $a$is 10% of $g$?

A gray kangaroo can bound across level ground with each jump

carrying it 10 m from the takeoff point. Typically the kangaroo

leaves the ground at a 20° angle. If this is so:

a. What is its takeoff speed?

b. What is its maximum height above the ground?

Answer Problems 3 through 5 by choosing one of the eight labeled acceleration vectors or selecting option $\text{I}:\overrightarrow{a}=\overrightarrow{0}$

5. At this instant, the particle is speeding up and curving downward. What is the direction of its acceleration?

For a projectile, which of the following quantities are constant during the flight: $x,y,r,v_x,v_y,v,a_x,a_y$? Which of these quantities are zero throughout the flight?

A ball is thrown toward a cliff of height h with a speed of 30 m/s and an angle of 60° above horizontal. It lands on the edge of the cliff 4.0 s later.
a. How high is the cliff?
b. What was the maximum height of the ball?
c. What is the ball’s impact speed?

A tennis player hits a ball 2.0 m above the ground. The ball

leaves his racquet with a speed of 20.0 m/s at an angle 5.0° above the horizontal. The horizontal distance to the net is 7.0 m, and the net is 1.0 m high. Does the ball clear the net? If so, by how much? If not, by how much does it miss?