Conceptual Physics

4.5 - Interactive problem: the monkey and the professor

The monkey at the right has a banana bazooka and plans to shoot a banana at a hungry physics professor. He has a glove to catch the banana. The professor is hanging from the tree, and the instant he sees the banana moving, he will drop from the tree in his eagerness to dine. Can you correctly aim the monkey’s bazooka so that the banana reaches the professor’s glove as he falls? Should you aim the shot above, below or directly at the professor? As long as the banana is fired fast enough to reach the professor before it hits the ground, does its initial speed matter? (Note: The simulation has a minimum speed so the banana will reach the professor when correctly aimed.) Give it a try in the interactive simulation to the right. No calculations are required to solve this problem. As you ponder your answer, consider the two key concepts of projectile motion: (1) all objects accelerate toward the ground at the same rate, and (2) the horizontal and vertical components of motion are independent. (The effect of air resistance is ignored.) Aim the banana by dragging the vector arrow at the end of the bazooka. You can increase the firing speed of the banana by making the arrow longer, and you can change the angle at which the banana is fired by moving the arrow up or down. Stretching out the vector makes it easy to aim the banana. To shoot the banana, press GO. Press RESET to try again. (Do not worry: We, too, value physics professors, so the professor will emerge unscathed.) If you have trouble with this problem, review the section on projectile motion.

4.6 - Interactive checkpoint: golfing

A golfer is on the edge of a 12.5 m

high bluff overlooking the eighteenth

hole, which is located 67.1 m from the

base of the bluff. She launches a

horizontal shot that lands in the hole

on the fly, and the gallery erupts into

cheers. How long was the ball in the

air? What was the ball’s horizontal

velocity? Take upward to be the

positiveydirection.

Answer:

t = s

vx = m/s

4.7 - Projectile motion: juggling

Juggling is a form of projectile motion in which the projectiles have initial velocities in both the vertical and horizontal dimensions. This motion takes more work to analyze than when a projectile’s initial vertical velocity is zero, as it was with the horizontally fired cannonball. Jugglers throw balls from one hand to the other and then back. To juggle multiple balls, the juggler repeats the same simple toss over and over. An experienced juggler’s ability to make this basic routine seem so effortless stems from the fact that the motion of each ball is identical. The balls always arrive at the same place for the catch, and in roughly the same amount of time.

A juggler throws each ball with an initial velocity that has both x and y components. Ignoring the effect of air resistance, the x component of the velocity remains constant as the ball moves in the air from one hand to another.

The initial y velocity is upward, which means it is a positive value. At all times, the ball accelerates downward at í9.80 m/s^2. This means the ball’s velocity decreases as it rises until it has a vertical velocity of zero. The ball then accelerates back toward the ground. When the ball plops down into the other hand, the magnitude of the y velocity will be the same as when the ball was tossed up, but its sign will be reversed. If the ball is thrown up with an initial y velocity of +5 m/s, it will land with a y velocity of í5 m/s. This symmetry is due to the constant rate of vertical

Projectile motion: y velocity

y velocity = zero at peak Initial y velocity equal but opposite to final y velocity