Motion in Space

1. Find the velocity, acceleration, and speed as functions of \(t\) for a particle with the given position function.   solution  \[\displaystyle{ \langle t^2, 2t, \ln t \rangle} \]

2. Given \( {\bf a}(t) = \sin t \, {\bf i} + 2 \cos t \, {\bf j} + 12t \, {\bf k}, \quad {\bf v}(0)=-{\bf k}, \quad {\bf r}(0) = {\bf j}+2{\bf k}\). Find the velocity and position vectors.   solution

3. A golf ball is hit in a horizontal direction off the top edge of a building that is 150 ft tall. How fast must the ball be launched to land 600 ft away?  solution

4. A projectile is fired with an initial speed of 200 m/s and angle of elevation 60°. Find (a) the range of the projectile,  (b) the maximum height reached, and (c) the speed at impact.  solution   Alt solution

5. A projectile is fired from a tank with initial speed 400 m/s. Find the angle of elevation that can be used to hit a target 3000 m away.   solution

6. A ball is thrown eastward (in the direction of the positive \(x\)-axis) into the air with an initial velocity of \(\langle 50, 0, 80 \rangle\) measured in feet per second. The spin of the ball results in a southward acceleration of \(4\) feet per second squared so the acceleration vector is \(\langle 0, -4, -32 \rangle\). Where does the ball land and with what speed (assume that the \(z\) coordinate measures elevation)?