Video Transcript
Free-Fall Conservation of Energy
If you throw a ball straight up in the air with an initial velocity of 10 m/s, how high will it go before it turns around and falls back to the ground?
One way to answer this question is by looking at how the energy of the ball changes as it goes up and then comes back down. There are two main kinds of energy that are important in this situation: kinetic and potential. Kinetic energy (K) is the energy of motion, with the equation (as you can see below) being:
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This is essentially saying that all moving objects have kinetic energy, and the amount of kinetic energy is proportional to both how fast the object is moving (velocity, v) and how heavy it is (mass, m).
In contrast to kinetic energy, an object can have gravitational potential energy whether it's moving or not. The amount of gravitational potential energy (UG) of the ball depends on how high it is above the ground (h) and its weight (W = mg, where g = 9.8 N/kg), with the equation, as you can see below, being:
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If no forces (other than gravity) act on the ball during its trip up and back down, then the ball's total mechanical energy (kinetic + potential) will not change. However, the energy of the ball can change forms. Initially, the ball has a lot of kinetic energy, because you've given it a pretty large upward velocity. In this case, the total energy is conserved because it doesn't change.
However, as the ball moves upward, it slows down as its initial kinetic energy is transformed into potential energy. Eventually, all of the initial kinetic energy will become potential energy, and the ball will stop momentarily. The ball has finally reached its highest point. After this, it'll turn around and fall back the ground as all the potential energy it had at the highest point is transformed back into kinetic energy as it falls to the ground.
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This change in energy can be represented using a bar chart that shows how much kinetic and potential energy the ball has at different times. Notice that the total energy is the same in both cases, but just after the ball is thrown, all its energy is kinetic. When it reaches the maximum height, all the energy has now been converted into potential energy.
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What Is Projectile Motion?
How would our analysis of the ball's motion change if it was not thrown straight up? What if the ball was thrown at an angle instead? When an object is moving through the air in both the horizontal and vertical directions, we call this projectile motion. Even though the motion is a little different than the motion of a ball that's thrown straight up, you can still determine how high the ball will go by examining how its energy changes.
Because gravity is still the only force acting on a projectile, the total energy will still not change. However, what's different this time is that the ball never reaches a point where it stops moving, even for an instant. When it gets to its maximum height, it's still moving forward, even though its vertical velocity is 0, so it still has some kinetic energy. Let's see what that would look like on a graph like the one below:
As you can see, when a ball is thrown at an angle, it never reaches a point where its kinetic energy is 0. However, some of its kinetic energy does become potential energy, and potential energy is at a maximum when the ball reaches its highest point.
Example Problem
If an archer fires a 0.20 kg arrow at an angle of 30 degrees above the horizontal and at an initial speed of 25 m/s, how high will the arrow go before it turns around and comes back to the ground?
To answer this question, you need to know several things. First, you need to know how much energy the arrow has at the beginning of its motion, just after it's fired. At this point, all of its energy is kinetic energy.
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As you can see, the origin of the coordinate system is the point where the arrow is launched, so it only has kinetic energy, as in y = 0, so the potential energy is 0. Using our formula, we get K = ½mv2, which, when we plug in our values is:
1/2(0.2 kg)(25 m/s)2
This gives us a kinetic energy of 62.5 J.
Next, think about what happens as the arrow goes up. This arrow is a projectile, and gravity is causing it to slow down in the vertical direction. However, no forces are acting on it in the horizontal direction, so its horizontal velocity won't change. When it reaches its maximum height, this is the point where the vertical velocity is 0. However, its horizontal velocity is still exactly the same as it was when the arrow was first launched from the bow. Calculate this horizontal velocity using the total initial velocity and the angle at which the arrow is shot.
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You can use this information to calculate how much kinetic energy the arrow still has at the top.
As you can see, at the maximum height, vertical velocity is 0, but horizontal velocity is still the same as it was at the beginning.
Finally, you can use conservation of energy to determine how much potential energy the arrow has at its maximum height, and then use that information to calculate the maximum height the arrow will reach.
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When we plug the kinetic energies we discovered into our other formula: 62.5 J + 0 = 46.9 J + Ug,f, which equals 15.6 J of kinetic energy; and finally, when plugging that value into our other equation, we get a yf of 8 m. And we're done!
Lesson Summary
Let's take a couple of moments to review what we've learned about the conservation of energy in projectile motion. When an object is moving through the air with velocity in both the horizontal and vertical directions, then the object is considered to be a projectile and its motion is called projectile motion. Since gravity is the only force acting on a projectile while it's in the air, then the total energy doesn't change at any point in its motion, although the energy can change forms from kinetic energy, K, which is the energy of motion, to potential energy, or from potential energy to kinetic energy.
