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Physics Calculator

Get detailed solutions to your math problems with our physics step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of physics. This solution was automatically generated by our smart calculator:

What do we already know? We know the values for acceleration ($a$), velocity ($v$), distance ($y$), height ($y_0$) and want to calculate the value of velocity ($v_0$)

According to the initial data we have about the problem, the following formula would be the most useful to find the unknown ($v_0$) that we are looking for. We need to solve the equation below for $v_0$

We substitute the data of the problem in the formula and proceed to simplify the equation

Multiply $-2$ times $9.81$

Multiply $-19.62$ times $3.2$

Calculate the power $0^2$

Rearrange the equation

We need to isolate the dependent variable $v_0$, we can do that by simultaneously subtracting $-62.784$ from both sides of the equation

 Intermediate steps

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{2}$

Divide $1$ by $2$

Simplify $\sqrt{v_0^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $0.5$

Multiply $2$ times $0.5$

Calculate the power $\sqrt{62.784}$

The complete answer is

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1.8: Solving Problems in Physics

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Learning Objectives

• Describe the process for developing a problem-solving strategy.
• Explain how to find the numerical solution to a problem.
• Summarize the process for assessing the significance of the numerical solution to a problem.

Problem-solving skills are clearly essential to success in a quantitative course in physics. More important, the ability to apply broad physical principles—usually represented by equations—to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life.

As you are probably well aware, a certain amount of creativity and insight is required to solve problems. No rigid procedure works every time. Creativity and insight grow with experience. With practice, the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and then progressing to the more difficult. After you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Although there is no simple step-by-step method that works for every problem, the following three-stage process facilitates problem solving and makes it more meaningful. The three stages are strategy, solution, and significance. This process is used in examples throughout the book. Here, we look at each stage of the process in turn.

Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:

• Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
• Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
• Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
• Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

The solution stage is when you do the math. Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units . That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.

Significance

After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:

• Check your units . If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
• Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
• Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these science that progresses.

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Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force.

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What is the 3-body problem, and is it really unsolvable?

The three-body problem is a physics conundrum that has boggled scientists since Isaac Newton's day. But what is it, why is it so hard to solve and is the sci-fi series of the same name really possible?

A rocket launch. Our nearest stellar neighbor. A Netflix show. All of these things have something in common: They must contend with the "three-body problem." But exactly what is this thorny physics conundrum?

The three-body problem describes a system containing three bodies that exert gravitational forces on one another. While it may sound simple, it's a notoriously tricky problem and "the first real worry of Newton," Billy Quarles , a planetary dynamicist at Valdosta State University in Georgia, told Live Science.

In a system of only two bodies, like a planet and a star, calculating how they'll move around each other is fairly straightforward: Most of the time, those two objects will orbit roughly in a circle around their center of mass, and they'll come back to where they started each time. But add a third body, like another star, and things get a lot more complicated. The third body attracts the two orbiting each other, pulling them out of their predictable paths .

The motion of the three bodies depends on their starting state — their positions, velocities and masses. If even one of those variables changes, the resulting motion could be completely different. 

"I think of it as if you're walking on a mountain ridge," Shane Ross , an applied mathematician at Virginia Tech, told Live Science. "With one small change, you could either fall to the right or you could fall to the left. Those are two very close initial positions, and they could lead to very different states."  

There aren't enough constraints on the motions of the bodies to solve the three-body problem with equations, Ross said. 

Related: Cosmic 'superbubbles' might be throwing entire galaxies into chaos, theoretical study hints

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But some solutions to the three-body problem have been found. For example, if the starting conditions are just right, three bodies of equal mass could chase one another in a figure-eight pattern. Such tidy solutions are the exception, however, when it comes to real systems in space.

Certain conditions can make the three-body problem easier to parse. Consider Tatooine , Luke Skywalker's fictional home world from "Star Wars" — a single planet orbiting two suns. Those two stars and the planet make up a three-body system. But if the planet is far enough away and orbiting both stars together, it's possible to simplify the problem. 

"When it's the Tatooine case, as long as you're far enough away from the central binary, then you think of this object as just being a really fat star," Quarles said. The planet doesn't exert much force on the stars because it's so much less massive, so the system becomes similar to the more easily solvable two-body problem. So far, scientists have found more than a dozen Tatooine-like exoplanets , Quarles told Live Science.

But often, the orbits of the three bodies never truly stabilize, and the three-body problem gets "solved" with a bang. The gravitational forces could cause two of the three bodies to collide, or they could fling one of the bodies out of the system forever — a possible source of "rogue planets" that don't orbit any star , Quarles said. In fact, three-body chaos may be so common in space that scientists estimate there may be 20 times as many rogue planets as there are stars in our galaxy.

When all else fails, scientists can use computers to approximate the motions of bodies in an individual three-body system. That makes it possible to predict the motion of a rocket launched into orbit around Earth, or to predict the fate of a planet in a system with multiple stars.

— 'Mathematically perfect' star system being investigated for potential alien technology

— How common are Tatooine worlds?

— Mathematicians find 12,000 new solutions to 'unsolvable' 3-body problem

With all this tumult, you might wonder if anything could survive on a planet like the one featured in Netflix's "3 Body Problem," which — spoiler alert — is trapped in a chaotic orbit around three stars in the Alpha Centauri system , our solar system 's nearest neighbor. 

"I don't think in that type of situation, that's a stable environment for life to evolve," Ross said. That's one aspect of the show that remains firmly in the realm of science fiction.

Skyler Ware is a freelance science journalist covering chemistry, biology, paleontology and Earth science. She was a 2023 AAAS Mass Media Science and Engineering Fellow at Science News. Her work has also appeared in Science News Explores, ZME Science and Chembites, among others. Skyler has a Ph.D. in chemistry from Caltech.

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Check Your Understanding

Answer: d = 1720 m

Answer: a = 8.10 m/s/s

Answers: d = 33.1 m and v f = 25.5 m/s

Answers: a = 11.2 m/s/s and d = 79.8 m

Answer: t = 1.29 s

Answers: a = 243 m/s/s

Answer: a = 0.712 m/s/s

Answer: d = 704 m

Answer: d = 28.6 m

Answer: v i = 7.17 m/s

Answer: v i = 5.03 m/s and hang time = 1.03 s (except for in sports commericals)

Answer: a = 1.62*10 5 m/s/s

Answer: d = 48.0 m

Answer: t = 8.69 s

Answer: a = -1.08*10^6 m/s/s

Answer: d = -57.0 m (57.0 meters deep) 

Answer: v i = 47.6 m/s

Answer: a = 2.86 m/s/s and t = 30. 8 s

Answer: a = 15.8 m/s/s

Answer: v i = 94.4 mi/hr

Solutions to Above Problems

t = 32.8 s v = 0 m/s d = (0 m/s)*(32.8 s)+ 0.5*(3.20 m/s 2 )*(32.8 s) 2 Return to Problem 1 t = 5.21 s v = 0 m/s 110 m = (0 m/s)*(5.21 s)+ 0.5*(a)*(5.21 s) 2 110 m = (13.57 s 2 )*a a = (110 m)/(13.57 s 2 ) a = 8.10 m/ s 2 Return to Problem 2 t = 2.6 s v = 0 m/s d = (0 m/s)*(2.60 s)+ 0.5*(-9.8 m/s 2 )*(2.60 s) 2 d = -33.1 m (- indicates direction) v f = v i + a*t v f = 0 + (-9.8 m/s 2 )*(2.60 s) v f = -25.5 m/s (- indicates direction) Return to Problem 3 v = 18.5 m/s v = 46.1 m/s t = 2.47 s a = (46.1 m/s - 18.5 m/s)/(2.47 s) a = 11.2 m/s 2 d = v i *t + 0.5*a*t 2 d = (18.5 m/s)*(2.47 s)+ 0.5*(11.2 m/s 2 )*(2.47 s) 2 d = 45.7 m + 34.1 m (Note: the d can also be calculated using the equation v f 2 = v i 2 + 2*a*d) Return to Problem 4 v = 0 m/s d = -1.40 m -1.40 m = (0 m/s)*(t)+ 0.5*(-1.67 m/s 2 )*(t) 2 -1.40 m = 0+ (-0.835 m/s 2 )*(t) 2 (-1.40 m)/(-0.835 m/s 2 ) = t 2 1.68 s 2 = t 2 Return to Problem 5 v = 0 m/s v = 444 m/s a = (444 m/s - 0 m/s)/(1.83 s) a = 243 m/s 2 d = (0 m/s)*(1.83 s)+ 0.5*(243 m/s 2 )*(1.83 s) 2 d = 0 m + 406 m Return to Problem 6 v = 0 m/s v = 7.10 m/s (7.10 m/s) 2 = (0 m/s) 2 + 2*(a)*(35.4 m) 50.4 m 2 /s 2 = (0 m/s) 2 + (70.8 m)*a (50.4 m 2 /s 2 )/(70.8 m) = a a = 0.712 m/s 2 Return to Problem 7 v = 0 m/s v = 65 m/s (65 m/s) 2 = (0 m/s) 2 + 2*(3 m/s 2 )*d 4225 m 2 /s 2 = (0 m/s) 2 + (6 m/s 2 )*d (4225 m 2 /s 2 )/(6 m/s 2 ) = d Return to Problem 8 v = 22.4 m/s v = 0 m/s d = (22.4 m/s + 0 m/s)/2 *2.55 s d = (11.2 m/s)*2.55 s Return to Problem 9 a = -9.8 m/s v = 0 m/s (0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(2.62 m) 0 m 2 /s 2 = v i 2 - 51.35 m 2 /s 2 51.35 m 2 /s 2 = v i 2 v i = 7.17 m/s Return to Problem 10 (0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(1.29 m) 0 m 2 /s 2 = v i 2 - 25.28 m 2 /s 2 25.28 m 2 /s 2 = v i 2 v i = 5.03 m/s To find hang time, find the time to the peak and then double it. 0 m/s = 5.03 m/s + (-9.8 m/s 2 )*t up -5.03 m/s = (-9.8 m/s 2 )*t up (-5.03 m/s)/(-9.8 m/s 2 ) = t up t up = 0.513 s hang time = 1.03 s Return to Problem 11 v = 0 m/s v = 521 m/s (521 m/s) 2 = (0 m/s) 2 + 2*(a)*(0.840 m) 271441 m 2 /s 2 = (0 m/s) 2 + (1.68 m)*a (271441 m 2 /s 2 )/(1.68 m) = a a = 1.62*10 5 m /s 2 Return to Problem 12 (NOTE: the time required to move to the peak of the trajectory is one-half the total hang time - 3.125 s.) First use:  v f  = v i  + a*t 0 m/s = v i  + (-9.8  m/s 2 )*(3.13 s) 0 m/s = v i  - 30.7 m/s v i  = 30.7 m/s  (30.674 m/s) Now use:  v f 2  = v i 2  + 2*a*d (0 m/s) 2  = (30.7 m/s) 2  + 2*(-9.8  m/s 2 )*(d) 0 m 2 /s 2  = (940 m 2 /s 2 ) + (-19.6  m/s 2 )*d -940  m 2 /s 2  = (-19.6  m/s 2 )*d (-940  m 2 /s 2 )/(-19.6  m/s 2 ) = d Return to Problem 13 v = 0 m/s d = -370 m -370 m = (0 m/s)*(t)+ 0.5*(-9.8 m/s 2 )*(t) 2 -370 m = 0+ (-4.9 m/s 2 )*(t) 2 (-370 m)/(-4.9 m/s 2 ) = t 2 75.5 s 2 = t 2 Return to Problem 14 v = 367 m/s v = 0 m/s (0 m/s) 2 = (367 m/s) 2 + 2*(a)*(0.0621 m) 0 m 2 /s 2 = (134689 m 2 /s 2 ) + (0.1242 m)*a -134689 m 2 /s 2 = (0.1242 m)*a (-134689 m 2 /s 2 )/(0.1242 m) = a a = -1.08*10 6 m /s 2 (The - sign indicates that the bullet slowed down.) Return to Problem 15 t = 3.41 s v = 0 m/s d = (0 m/s)*(3.41 s)+ 0.5*(-9.8 m/s 2 )*(3.41 s) 2 d = 0 m+ 0.5*(-9.8 m/s 2 )*(11.63 s 2 ) d = -57.0 m (NOTE: the - sign indicates direction) Return to Problem 16 a = -3.90 m/s v = 0 m/s (0 m/s) 2 = v i 2 + 2*(- 3.90 m/s 2 )*(290 m) 0 m 2 /s 2 = v i 2 - 2262 m 2 /s 2 2262 m 2 /s 2 = v i 2 v i = 47.6 m /s Return to Problem 17 v = 0 m/s v = 88.3 m/s ( 88.3 m/s) 2 = (0 m/s) 2 + 2*(a)*(1365 m) 7797 m 2 /s 2 = (0 m 2 /s 2 ) + (2730 m)*a 7797 m 2 /s 2 = (2730 m)*a (7797 m 2 /s 2 )/(2730 m) = a a = 2.86 m/s 2 88.3 m/s = 0 m/s + (2.86 m/s 2 )*t (88.3 m/s)/(2.86 m/s 2 ) = t t = 30. 8 s Return to Problem 18 v = 0 m/s v = m/s ( 112 m/s) 2 = (0 m/s) 2 + 2*(a)*(398 m) 12544 m 2 /s 2 = 0 m 2 /s 2 + (796 m)*a 12544 m 2 /s 2 = (796 m)*a (12544 m 2 /s 2 )/(796 m) = a a = 15.8 m/s 2 Return to Problem 19 v f 2 = v i 2 + 2*a*d (0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(91.5 m) 0 m 2 /s 2 = v i 2 - 1793 m 2 /s 2 1793 m 2 /s 2 = v i 2 v i = 42.3 m/s Now convert from m/s to mi/hr: v i = 42.3 m/s * (2.23 mi/hr)/(1 m/s) v i = 94.4 mi/hr Return to Problem 20 403 IMAGES Best Physics Apps for Students Tips for Solving Physics Problems Watch How To Solve Any Physics Problem ? 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