In the What If? section of Example 4.5, it was claimed that the maximum range of a ski jumper occurs for a launch angle \(\theta\) given by \(\theta\) = \(45^{\circ}\) - \(\frac{\phi}{2}\) where \(\phi\) is the angle the hill makes with the horizontal in Figure 4.15. Prove this claim by deriving the equation above.

A ball on the end of a string is whirled around in a horizontal circle of radius 0.300 m. The plane of the circle is 1.20 m above the ground. The string breaks and the ball lands 2.00 m (horizontally) away from the point on the ground directly beneath the ball’s location when the string breaks. Find the radial acceleration of the ball during its circular motion.

Why is the following situation impossible? A normally proportioned adult walks briskly along a straight line in the +x direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing. His right hand holds a ball at his side a distance h above the floor. When the ball passes above a point marked as x = 0 on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand. The ball strikes the ground for the first time at position x = 7.00h.

A particle starts from the origin with velocity \(5 \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}\) at t = 0 and moves in the xy plane with a varying acceleration given by \(\overrightarrow{\mathbf{a}}\) = \((6 \sqrt{t} \hat{\mathbf{j}})\), where \(\overrightarrow{\mathbf{a}}\) is in meters per second squared and t is in seconds. (a) Determine the velocity of the particle as a function of time. (b) Determine the position of the particle as a function of time.

Lisa in her Lamborghini accelerates at \((3.00 \hat{\mathbf{i}}-2.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^2\), while Jill in her Jaguar accelerates at \((1.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^2\). They both start from rest at the origin. After 5.00 s, (a) what is Lisa's speed with respect to Jill, (b) how far apart are they, and (c) what is Lisa's acceleration relative to Jill?

A boy throws a stone horizontally from the top of a cliff of height h toward the ocean below. The stone strikes the ocean at distance d from the base of the cliff. In terms of h, d, and g, find expressions for (a) the time t at which the stone lands in the ocean, (b) the initial speed of the stone, (c) the speed of the stone immediately before it reaches the ocean, and (d) the direction of the stone's velocity immediately before it reaches the ocean.

Why is the following situation impossible? Albert Pujols hits a home run so that the baseball just clears the top row of bleachers, 24.0 m high, located 130 m from home plate. The ball is hit at 41.7 m/s at an angle of \(35.0^{\circ}\) to the horizontal, and air resistance is negligible.

As some molten metal splashes, one droplet flies off to the east with initial velocity \(v_i\) at angle \(\theta_i\) above the horizontal, and another droplet flies off to the west with the same speed at the same angle above the horizontal as shown in Figure P4.40. In terms of \(v_i\) and \(\theta_i\), find the distance between the two droplets as a function of time.

An astronaut on the surface of the Moon fires a cannon to launch an experiment package, which leaves the barrel moving horizontally. Assume the free-fall acceleration on the Moon is one-sixth of that on the Earth. (a) What must the muzzle speed of the package be so that it travels completely around the Moon and returns to its original location? (b) What time interval does this trip around the Moon require?

A pendulum with a cord of length r = 1.00 m swings in a vertical plane (Fig. P4.42). When the pendulum is in the two horizontal positions \(\theta\) = \(90.0^{\circ}\) and \(\theta\) = \(270^{\circ}\), its speed is 5.00 m/s. Find the magnitude of (a) the radial acceleration and (b) the tangential acceleration for these positions. (c) Draw vector diagrams to determine the direction of the total acceleration for these two positions. (d) Calculate the magnitude and direction of the total acceleration at these two positions.

A spring cannon is located at the edge of a table that is 1.20 m above the floor. A steel ball is launched from the cannon with speed \(v_i\) at \(35.0^{\circ}\) above the horizontal. (a) Find the horizontal position of the ball as a function of \(v_i\) at the instant it lands on the floor. We write this function as \(x\left(v_i\right)\). Evaluate x for (b) \(v_i\) = 0.100 m/s and for (c) \(v_i\) = 100 m/s. (d) Assume \(v_i\) is close to but not equal to zero. Show that one term in the answer to part (a) dominates so that the function \(x\left(v_i\right)\) reduces to a simpler form. (e) If \(v_i\) is very large, what is the approximate form of \(x\left(v_i\right)\)? (f) Describe the overall shape of the graph of the function \(x\left(v_i\right)\).

A projectile is launched from the point (x = 0, y = 0), with velocity \((12.0 \hat{\mathbf{i}}+49.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}\), at t = 0. (a) Make a table listing the projectile's distance \(|\overrightarrow{\mathbf{r}}|\) from the origin at the end of each second thereafter, for \(0 \leq t \leq\) 10 s. Tabulating the x and y coordinates and the components of velocity \(v_x\) and \(v_y\) will also be useful. (b) Notice that the projectile's distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the velocity. Suggestion: Argue that if \(\overrightarrow{\mathbf{v}}\) is not perpendicular to \(\overrightarrow{\mathbf{r}}\), then \(|\overrightarrow{\mathbf{r}}|\) must be increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) Explain your method for solving part (c).

A fisherman sets out upstream on a river. His small boat, powered by an outboard motor, travels at a constant speed v in still water. The water flows at a lower constant speed \(v_{w^v}\). The fisherman has traveled upstream for 2.00 km when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 15.0 min. At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as he returns to his starting point. How fast is the river flowing? Solve this problem in two ways. (a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed v - \(v_w\) and downstream at v + \(v_w\) (b) A second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets.

An outfielder throws a baseball to his catcher in an attempt to throw out a runner at home plate. The ball bounces once before reaching the catcher. Assume the angle at which the bounced ball leaves the ground is the same as the angle at which the outfielder threw it as shown in Figure P4.46, but that the ball’s speed after the bounce is one-half of what it was before the bounce. (a) Assume the ball is always thrown with the same initial speed and ignore air resistance. 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 time interval for the one-bounce throw to the flight time for the no-bounce throw.

Do not hurt yourself; do not strike your hand against anything. Within these limitations, describe what you do to give your hand a large acceleration. Compute an order-of-magnitude estimate of this acceleration, stating the quantities you measure or estimate and their values.

You are on the Pirates of the Caribbean attraction in the Magic Kingdom at Disney World. Your boat rides through a pirate battle, in which cannons on a ship and in a fort are firing at each other. While you are aware that the splashes in the water do not represent actual cannonballs, you begin to wonder about such battles in the days of the pirates. Suppose the fort and the ship are separated by 75.0 m. You see that the cannons in the fort are aimed so that their cannonballs would be fired horizontally from a height of 7.00 m above the water. (a) You wonder at what speed they must be fired in order to hit the ship before falling in the water. (b) Then, you think about the sludge that must build up inside the barrel of a cannon. This sludge should slow down the cannonballs. A question occurs in your mind: if the cannonballs can be fired at only 50.0% of the speed found earlier, is it possible to fire them upward at some angle to the horizontal so that they would reach the ship?

A skier leaves the ramp of a ski jump with a velocity of v = 10.0 m/s at \(\theta\) = \(15.0^{\circ}\) above the horizontal as shown in Figure P4.49 (page 94). The slope where she will land is inclined downward at \(\phi\) = \(50.0^{\circ}\), and air resistance is negligible. Find (a) the distance from the end of the ramp to where the jumper lands and (b) her velocity components just before the landing. (c) Explain how you think the results might be affected if air resistance were included.

A projectile is fired up an incline (incline angle \(\phi\)) with an initial speed \(v_i\) at an angle \(\theta_i\) with respect to the horizontal \(\left(\theta_i>\phi\right)\) as shown in Figure P4.50. (a) Show that the projectile travels a distance d up the incline, where d = \(\frac{2 v_i^2 \cos \theta_i \sin \left(\theta_i-\phi\right)}{g \cos ^2 \phi}\) (b) For what value of \(\theta_i\) is d a maximum, and what is that maximum value?

Two swimmers, Chris and Sarah, start together at the same point on the bank of a wide stream that flows with a speed v. Both move at the same speed c (where c > v) relative to the water. Chris swims downstream a distance L and then upstream the same distance. Sarah swims so that her motion relative to the Earth is perpendicular to the banks of the stream. She swims the distance L and then back the same distance, with both swimmers returning to the starting point. In terms of L, c, and v, find the time intervals required (a) for Chris's round trip and (b) for Sarah's round trip. (c) Explain which swimmer returns first.

In the What If? section of Example 4.5, it was claimed that the maximum range of a ski jumper occurs for a launch angle \(\theta\) given by \(\theta\) = \(45^{\circ}\) - \(\frac{\phi}{2}\) where \(\phi\) is the angle the hill makes with the horizontal in Figure 4.15. Prove this claim by deriving the equation above.

A fireworks rocket explodes at height h, the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed v. Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal.