When freeways were first built in the United States in the $1950$s, entrance ramps were often too short for an entering car to safely merge into existing traffic. Consider an aggressive car acceleration of $a=4.0 \mathrm{~m} / \mathrm{s}^2$ and an initial car speed of $v_0=25 \mathrm{mi} / \mathrm{h}$ as the car enters the entrance ramp of a freeway where other cars are moving at $55 \mathrm{mi} / \mathrm{h}$. $(a)$ If the ramp has a length of $d=40 \mathrm{yd}$, what is the car's speed $v$ in miles per hour as it attempts to merge? $(b)$ What is the minimum length $d$ in yards needed for the car's speed to match the speed of the other cars?

The running event known as the $100 \mathrm{~m}$ dash consists of three stages. In the first, the runner accelerates to the maximum speed, which usually occurs at the $50 \mathrm{~m}$ to $70 \mathrm{~m}$ mark. That speed is then maintained until the last $10 \mathrm{~m}$, when the runner slows. Consider three parts of the record-setting run by Usain Bolt in the $2008$ Olympics: $(a)$ from $10 \mathrm{~m}$ to $20 \mathrm{~m}$, elapsed time of $1.02 \mathrm{~s}$, $(b)$ from $50 \mathrm{~m}$ to $60 \mathrm{~m}$, elapsed time of $0.82 \mathrm{~s}$, and $(c)$ from $90 \mathrm{~m}$ to $100 \mathrm{~m}$, elapsed time of $0.90 \mathrm{~s}$. What was the average velocity for each part?

The running event known as the $100 \mathrm{~m}$ dash consists of three stages. In the first, the runner accelerates to the maximum speed, which usually occurs at the $50 \mathrm{~m}$ to $70 \mathrm{~m}$ mark. That speed is then maintained until the last $10 \mathrm{~m}$, when the runner slows. Consider three parts of the record-setting run by Usain Bolt in the 2008 Olympics: (a) from $10 \mathrm{~m}$ to $20 \mathrm{~m}$, elapsed time of $1.02 \mathrm{~s}$, (b) from $50 \mathrm{~m}$ to $60 \mathrm{~m}$, elapsed time of $0.82 \mathrm{~s}$, and (c) from $90 \mathrm{~m}$ to $100 \mathrm{~m}$, elapsed time of $0.90 \mathrm{~s}$. What was the average velocity for each part?

When startled, an armadillo will leap upward. Suppose it rises 0.544 m in the first 0.200 s. What is its initial speed as it leaves the ground?

Figure $2.36$ a gives an overhead view of three cars with the same length $L=4.50 \mathrm{~m}$. Cars $A$ and $B$ are moving at $v_A=22.0 \mathrm{~m} / \mathrm{s}(49 \mathrm{mi} / \mathrm{h})$ along the right-hand lane of a long, straight road with two lanes in each direction, and car $C$ is moving along the passing lane at $v_C=27.0 \mathrm{~m} / \mathrm{s}(60 \mathrm{mi} / \mathrm{h})$ at initial distance $d$ behind $B$. Car $B$ is autonomous and is equipped with a computer control system using radar to detect the speeds and distances of the other two cars. At time $t=0$, the front of $\operatorname{car} B$ is $3.00 L$ behind the rear of $\operatorname{car} A$, which is at $x_{A 1}=0$ on the $x$ axis. We want $B$ to pull into the passing lane, speed up and pass $A$, and then pull back into the right-hand lane, $3.00 \mathrm{~L}$ in front of $A$ and at the initial speed. The computer control system will allow $15.0 \mathrm{~s}$ for the maneuver and only if the front of $C$ will be no closer than $3.00 L$ behind $A$ at the end of the maneuver, as in Fig. 2.36b. What is the least value of $d$ that the system will allow?

In professional baseball, the pitching distance of $60$ feet $6$ inches is the distance from the front of the pitcher's plate (or rubber) to the rear of the home plate. $(a)$ Assuming that a $95 \mathrm{mi} / \mathrm{h}$ fastball travels that full distance horizontally, what is its flight time, which is the time a batter must judge if the ball is "hittable" and then swing the bat? $(b)$ Research indicates that even an elite batter cannot track the ball for the full flight and yet many players have described seeing the ball-bat collision. One explanation is that the eyes track the ball in the early part of the flight and then undergo a predictive saccade in which they jump to an anticipated point later in the flight. A saccade suppresses vision for $20 \mathrm{~ms}$. How far in feet does the fastball travel during that interval of no vision?

A bolt is dropped from a bridge under construction, falling 90 m to the valley below the bridge. What is its speed when it reaches the valley beneath the bridge?

A room has dimensions 3.00 m (height) × 3.70 m × 4.30 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) Could the length of its path be less than this magnitude? (c) Greater? (d) Equal? (e) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. (f) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

If the position of a particle is given by $x = 20t - 5t^3$, where $x$ is in meters and $t$ is in seconds, when, if ever, is the particle’s velocity zero?

If a particle’s position is given by $x = 4- 12t + 3t^2$ (where t is in seconds and x is in meters), what is its velocity at t = 1 s?

During liftoff, a hot-air balloon accelerates upward at a rate of $3.0\ \mathrm{m/ s^2}$ . The balloonist drops an object over the side of the gondola when the speed is $15\ \mathrm{m/ s}$. (a) What is the object's acceleration after it is released (relative to the ground)? (b) How long does it take to hit the ground?

The smallest time interval defined in physics is the Planck time $t_{\mathrm{p}}=5.39 \times 10^{-44} \mathrm{~s}$, which is the time required for light to travel across a certain length in a vacuum. The universe began with the big bang $13.772$ billion years ago. What is the number of Planck times since that beginning?

A fortnight is a charming English measure of time equal to 2.0 weeks (the word is a contraction of “fourteen nights”).That is a nice amount of time in pleasant company but perhaps a painful string of micro seconds in unpleasant company. How many microseconds are in a fortnight?

A basketball player grabbing a rebound jump s 78.0 cm vertically, how much total time (ascent and descent) does the player spend (a) in the top 15.0 cm of this jump and (b) in the bottom 15.0 cm? Do your results explain why such players seem to hang in the air at the top of a jump?