In Figure 2.11b, the area under the velocity–time graph and between the vertical axis and time t (vertical dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. (a) Compute their areas. (b) Explain how the sum of the two areas compares with the expression on the right-hand side of Equation 2.16.

In Figure 2.11b, the area under the velocity–time graph and between the vertical axis and time t (vertical dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. (a) Compute their areas. (b) Explain how the sum of the two areas compares with the expression on the right-hand side of Equation 2.16.

The froghopper Philaenus spumarius is supposedly the best jumper in the animal kingdom. To start a jump, this insect can accelerate at 4.00 km/\(s^2\) over a distance of 2.00 mm as it straightens its specially adapted “jumping legs.” Assume the acceleration is constant. (a) Find the upward velocity with which the insect takes off. (b) In what time interval does it reach this velocity? (c) How high would the insect jump if air resistance were negligible? The actual height it reaches is about 70 cm, so air resistance must be a noticeable force on the leaping froghopper.

A woman is reported to have fallen 144 ft from the 17th floor of a building, landing on a metal ventilator box that she crushed to a depth of 18.0 in. She suffered only minor injuries. Ignoring air resistance, calculate (a) the speed of the woman just before she collided with the ventilator and (b) her average acceleration while in contact with the box. (c) Modeling her acceleration as constant, calculate the time interval it took to crush the box.

At t = 0, one athlete in a race running on a long, straight track with a constant speed \(v_1\) is a distance \(d_1\) behind a second athlete running with a constant speed \(v_2\). (a) Under what circumstances is the first athlete able to overtake the second athlete? (b) Find the time t at which the first athlete overtakes the second athlete, in terms of \(d_1\), \(v_1\), and \(v_2\). (c) At what minimum distance \(d_2\) from the leading athlete must the finish line be located so that the trailing athlete can at least tie for first place? Express \(d_2\) in terms of \(d_1\), \(v_1\), and \(v_2\) by using the result of part (b).

Why is the following situation impossible? A freight train is lumbering along at a constant speed of 16.0 m/s. Behind the freight train on the same track is a passenger train traveling in the same direction at 40.0 m/s. When the front of the passenger train is 58.5 m from the back of the freight train, the engineer on the passenger train recognizes the danger and hits the brakes of his train, causing the train to move with acceleration -3.00 m/\(s^2\). Because of the engineer's action, the trains do not collide.

Hannah tests her new sports car by racing with Sam, an experienced racer. Both start from rest, but Hannah leaves the starting line 1.00 s after Sam does. Sam moves with a constant acceleration of 3.50 m/\(s^2\), while Hannah maintains an acceleration of 4.90 m/\(s^2\). Find (a) the time at which Hannah overtakes Sam, (b) the distance she travels before she catches him, and (c) the speeds of both cars at the instant Hannah overtakes Sam.

Two objects, A and B, are connected by hinges to a rigid rod that has a length L. The objects slide along perpendicular guide rails as shown in Figure P2.40. Assume object A slides to the left with a constant speed v. (a) Find the velocity \(v_{\mathrm{B}}\) of object B as a function of the angle \(\theta\). (b) Describe \(v_{\mathrm{B}}\) relative to v. Is \(v_{\mathrm{B}}\) always smaller than v, larger than v, or the same as v, or does it have some other relationship?

Lisa rushes down onto a subway platform to find her train already departing. She stops and watches the cars go by. Each car is 8.60 m long. The first moves past her in 1.50 s and the second in 1.10 s. Find the constant acceleration of the train.

Two thin rods are fastened to the inside of a circular ring as shown in Figure P2.42. One rod of length D is vertical, and the other of length L makes an angle \(\theta\) with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods. (a) If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first. (b) Find an expression for the time interval required for the red bead to fall from point ? to point ? in terms of g and D. (c) Find an expression for the time interval required for the blue bead to slide from point ? to point ? in terms of g, L, and \(\theta\). (d) Show that the two time intervals found in parts (b) and (c) are equal. Hint: What is the angle between the chords of the circle ? ? and ? ?? (e) Do these results surprise you? Was your intuitive guess in part (a) correct? This problem was inspired by an article by Thomas B. Greenslade, Jr., “Galileo’s Paradox,” Phys. Teach. 46, 294 (May 2008).

In a women’s 100-m race, accelerating uniformly, Laura takes 2.00 s and Healan 3.00 s to attain their maximum speeds, which they each maintain for the rest of the race. They cross the finish line simultaneously, both setting a world record of 10.4 s. (a) What is the acceleration of each sprinter? (b) What are their respective maximum speeds? (c) Which sprinter is ahead at the 6.00-s mark, and by how much? (d) What is the maximum distance by which Healan is behind Laura, and at what time does that occur?

Review. You are sitting in your car at rest at a traffic light with a bicyclist at rest next to you in the adjoining bicycle lane. As soon as the traffic light turns green, your car speeds up from rest to 50.0 mi/h with constant acceleration 9.00 mi/h/s and thereafter moves with a constant speed of 50.0 mi/h. At the same time, the cyclist speeds up from rest to 20.0 mi/h with constant acceleration 13.0 mi/h/s and thereafter moves with a constant speed of 20.0 mi/h. (a) For what time interval after the light turned green is the bicycle ahead of your car? (b) What is the maximum distance by which the bicycle leads your car during this time interval?