Question

must show complete work!!

1. a plane flying horizontally at 350 m/s releases a package at an altitude of 1500 m. (a) how long will the package take to hit the ground? (b) how far will it move horizontally while falling?
2. a stone is thrown horizontally at a speed of 12 m/s from the top of a cliff 80 m high. (a) how long does it take the stone to reach the bottom of the cliff? (b) how far from the base of the cliff does the stone strike the ground?
3. a steel ball with a constant velocity of 0.8 m/s rolls off the edge of a table. the table is 0.95 m high. (a) how long is the ball in the air? (b) how far from the edge of the table does the ball land?

Explanation:

Step1: Analyze vertical motion for time (for all parts (a))

The vertical motion of the package, stone, and ball is a free - fall motion. The vertical displacement formula is \(y = v_{0y}t+\frac{1}{2}gt^{2}\). Since the initial vertical velocity \(v_{0y} = 0\) (thrown or released horizontally), the formula simplifies to \(y=\frac{1}{2}gt^{2}\), and we can solve for \(t\) as \(t=\sqrt{\frac{2y}{g}}\). For \(g = 9.8\ m/s^{2}\).

• Problem 1(a):

\(y = 1500\ m\), then \(t=\sqrt{\frac{2\times1500}{9.8}}\approx\sqrt{\frac{3000}{9.8}}\approx17.5\ s\)

• Problem 2(a):

\(y = 80\ m\), then \(t=\sqrt{\frac{2\times80}{9.8}}=\sqrt{\frac{160}{9.8}}\approx4.04\ s\)

• Problem 3(a):

\(y = 0.95\ m\), then \(t=\sqrt{\frac{2\times0.95}{9.8}}=\sqrt{\frac{1.9}{9.8}}\approx0.44\ s\)

Step2: Analyze horizontal motion for horizontal distance (for all parts (b))

The horizontal motion is a uniform - motion with constant velocity since there is no horizontal acceleration (\(a_x = 0\)). The horizontal distance formula is \(x=v_{0x}t\)

• Problem 1(b):

\(v_{0x}=350\ m/s\), \(t\approx17.5\ s\), then \(x = 350\times17.5=6125\ m\)

• Problem 2(b):

\(v_{0x}=12\ m/s\), \(t\approx4.04\ s\), then \(x = 12\times4.04 = 48.48\ m\)

• Problem 3(b):

\(v_{0x}=0.8\ m/s\), \(t\approx0.44\ s\), then \(x=0.8\times0.44 = 0.35\ m\)

Answer:

1. (a) \(t\approx17.5\ s\) (b) \(x = 6125\ m\)
2. (a) \(t\approx4.04\ s\) (b) \(x\approx48.48\ m\)
3. (a) \(t\approx0.44\ s\) (b) \(x\approx0.35\ m\)