Problem: Article: Higher-level speed calculations can get confusing because mathematicians and scientists use different definitions for "speed" and "velocity". A velocity has two components: a magnitude and a direction. The magnitude is equal to the object's speed. A change in the direction will cause a change in the velocity, but not in the speed. For example, let's say that there are two cars moving in opposite directions. Both cars' speedometers read 50 km/hr, so they both have the same speed. However, since they are moving apart from each other, we say that one car has a velocity of -50 km/hr and one has a velocity of 50 km/hr. Just as you can calculate instantaneous speed, you can also calculate instantaneous velocity. Objects can have velocities with a negative magnitude (if they are moving in a negative direction relative to something else). However, there's no such thing as a negative speed, so in these cases the absolute value of the magnitude gives the object's speed. For this reason, in the example problem above, both cars have a speed of 50 km/hr. If you have a function s(t) that gives you the position of an object with regards to time, the derivative of s(t) will give you its velocity with regards to time. Just plug a time value into this equation for the variable t (or whatever the time value is) to get the velocity at this given time. From here, finding the speed is easy. For example, let's say that an object's position in meters is given with the equation 3t2 + t - 4 where t = time in seconds. We want to know what the speed of the object is at t = 4 seconds. In this case, we can solve like this: 3t2 + t - 4 s'(t) = 2 3t + 1 s'(t) = 6t + 1 Now, we plug in t = 4: s'(t) = 6(4) + 1 = 24 + 1 = 25 meters/second. This is technically a velocity measurement, but since it's positive and direction is not mentioned in the problem, we can essentially use it for speed. Acceleration is a way of measuring the change in an object's velocity over time. This topic is a little too complex to explain fully in this article. However, it's useful to note that when you have a function a(t) that gives acceleration with regards to time, the integral of a(t) will give you velocity with regards to time. Note that it's helpful to know the object's initial velocity so that you can define the constant that results from an indefinite integral. For example, let's say that an object has a constant acceleration (in m/s2 given by a(t) = -30. Let's also say that it has an initial velocity of 10 m/s. We need to find its speed at t = 12 s. In this case, we can solve like this: a(t) = -30 v(t)= a(t)dt = -30dt = -30t + C To find C, we'll solve v(t) for t = 0. Remember that the object's initial velocity is 10 m/s. v(0) = 10 = -30(0) + C 10 = C, so v(t) = -30t + 10 Now, we can just plug in t = 12 seconds. v(12) = -30(12) + 10 = -360 + 10 = -350. Since speed is the absolute value of velocity, the object's speed is 350 meters/second.
Summary: Understand that speed is defined as the magnitude of velocity. Use absolute values for negative velocities. Take the derivative of a position function. Take the integral of an acceleration function.

Problem: Article: Fire up the grill and let it begin warming up as you finish getting the sprouts ready. A moderate temperature (somewhere around 350425F (177218C)) will sear the outside of the sprouts nicely. Be sure to light both burners or sides of the grill so that the heat stays consistent. Scrub the grill grate thoroughly with a wire brush prior to cooking to remove any leftover bits of charred food that might influence the flavor of the brussels sprouts. If you're cooking on a charcoal grill, go easy on the lighter fluid. The fumes can easily overpower mild ingredients. Lay out the sprouts across the grate with the stem-ends facing down. That way, they'll be taking the most direct heat and you won't have to worry about them falling apart. Spread the skewers out over the grill surface so they'll be easier to turn and remove once they're done cooking. If you situate the skewers so that they're running parallel to the grate, there's a chance they could slip down through the openings as they cook down. Grilling the sprouts directly on the grate results in a bolder flavor than wrapping them in foil or cooking them on one of the upper racks. Turn the skewers over after the first 5 minutes, when the sprouts have turned a deep green color and begun to lightly brown around the edges. Especially large or tough sprouts may require as long as 10 minutes per side to heat through. Exact cooking times may vary, depending on the size, texture, and number of the sprouts. Keep an eye on your brussels sprouts as they grill and use your best judgment to determine when they're done. As the sprouts cook, the coating of oil on the outside will cause them to caramelize, making them nice and crispy.
Summary: Preheat the grill to a medium-high heat. Throw the skewers on the grill. Cover and grill for about 5-10 minutes on each side.

Problem: Article: While summer is a great time to visit Yosemite Valley, everyone else thinks so, too. April to October is the main tourist season, and during this time, the park will be stuffed with visitors. If you want your visit to be more low-key, try going during another part of the year. Parts of the park do close during winter, but Yosemite Valley remains open. Most people don't like getting up early on vacation, but if you do, you can beat some of the crowds. While there, plan to get to the park before 9 am to avoid traffic. Alternatively, you can get there a bit later, say after 5 pm, when the traffic starts letting up again. Yosemite has a wide range of activities to pursue. You can hike, backpack, and fish, just to name a few. You can also drive around the park, picnic, stargaze, or even ride a horse. Enjoy water sports, rock climbing, and winter sports if you're more adventurous. Deciding ahead of time gives you a chance to plan out your schedule. Plus, you'll likely want to bring stuff with you for certain activities, such as hiking boots if you plan on doing extensive hiking. Yosemite has a wide range of educational resources for you to enjoy. You can visit places like the Visitor's Center, the Ansel Adam's Gallery, and the research library. You can also see the Yosemite Museum and the Yosemite Chapel.
Summary:
Visit in the off-season for a low-key vibe. Plan to get up early in the morning. Decide on activities. Explore the educational and recreational resources.