Calculating Kinetic Energy and Average Force in Projectile Throws

In summary: When you throw an object, it has a certain initial velocity. After you release it, the object will keep moving in the direction it was thrown (ignoring air resistance). However, as the object moves, Earth's gravity will pull down on it and cause it to accelerate downward. This downward acceleration causes the object to change direction; the object's velocity will no longer be horizontal, it will be at an angle. The object will continue to accelerate downward until it hits the ground. The object's velocity will be constantly changing as it moves under the influence of gravity.In summary, projectile motion can be resolved into two perpendicular components: horizontal and vertical. By using the law of conservation of mechanical energy, the minimum kinetic energy required for an object
  • #1
John H
36
0

Homework Statement


The masses of the javelin, discus, and shot are 0.8kg, 2kg, 7.2kg, respectively, and record throws in the track events using these objects are about 89 m, 69 m, 21 m. respectively. Neglecting air resistance, (a) calculate the minimum kinetic energies that would produce such throws, and (b) estimate the average force exerted on each object during the throw, assuming that force acts over a distance of 2m.


Homework Equations


W= [tex]\Delta[/tex]Ek
W= [tex]\Delta[/tex]Eg
w = F [tex]\times[/tex] [tex]\Delta[/tex]d
Eg= mg[tex]\Delta[/tex]h
Ek = 1/2mv^2

The Attempt at a Solution


I attempted to use the law of conservation of mechanical energy, ignoring any effects a non conservative force such as air resistance would have on the system, but am unable to find initial velocity due to the fact that i do not have the height of the projectiles at their maximum point, and i see no way to mathematically remove them from my calculations
 
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  • #2
Write down the equation for the projectile motion. The bodies have horizontal and a vertical velocity components (vx, vy respectively). How these components vary with time? How is related vx a to the length of the shot? How is the time of flight related to the initial vertical velocity?

ehild
 
  • #3
Sorry, but I have never even been though projectile motion, this is a grade 11 physics problem.
 
  • #4
Sorry, but I have never even been though projectile motion, this is a grade 11 physics problem.

Projectile Motion is there in Grade 11. You need to check again.
 
  • #5
snshusat161 said:
Projectile Motion is there in Grade 11. You need to check again.

Okay, I actually want to learn projectile motion, but in our cricilum in Canada we don't do that until grade 12. We did kinematics and dynamics, followed by work, power, and energy
 
  • #6
Okay, I actually want to learn projectile motion, but in our cricilum in Canada we don't do that until grade 12. We did kinematics and dynamics, followed by work, power, and energy

This is very easy topic. Do you know about angles and how to resolve them in two perpendicular components?
 
  • #7
I know about angles, and i did a little bit of reading into vector components my self, again we do components in grade 12. In this question where not given time, or height, the only possible way I could think of solvign the problem is by persuming the initial velocity, and the angle its being thrown at.
 
  • #8
I know about angles, and i did a little bit of reading into vector components my self, again we do components in grade 12. In this question where not given time, or height, the only possible way I could think of solvign the problem is by persuming the initial velocity, and the angle its being thrown at.

I've asked this because I thought telling you something about projectile motion. One think I'm thinking, as kinetic energy and work both are independent of the path taken so there is no need to know projectile motion in this problem. This is only my thinking and I'm not very good in Physics. What do you think about it.
 
  • #9
snshusat161 said:
I've asked this because I thought telling you something about projectile motion. One think I'm thinking, as kinetic energy and work both are independent of the path taken so there is no need to know projectile motion in this problem. This is only my thinking and I'm not very good in Physics. What do you think about it.

But then wouldn't we have to persume the amount of force the athelete applied to the ball.
 
  • #10
The motion of these objects (projectiles) can be resolved into a vertical and a horizontal motion, each with an initial velocity. The vertical motion is under the influence of gravity, with acceleration of g. The horizontal velocity [itex]v_x[/itex] is constant. If the object is in air for T time, it travels [itex]D=v_x\cdot T[/itex] horizontal distance. During the same time, it rises to the maximum height and falls back. The rise time is equal to the time of falling down: T/2. If the initial vertical velocity is [itex]v_y[/itex], it will decrease uniformly with rate g and becomes zero at the highest point, in T/2 time:
[itex]v_y=g \cdot T/2[/itex]

The initial kinetic energy is

[tex]KE=0.5 m (v_x^2 +v_y^2 )= 0.5 m (0.25 g^2\cdot T^2+D^2/T^2) [/tex]

and you want it to be minimum.

Local monimum can exist if the derivative with respect to T is zero, that means

[tex]0.5 \cdot g^2\cdot T-2\cdot D^2/T^3 = 0 \rightarrow T^4=4D^2/g^2 \rightarrow T=\sqrt{2D/g}[/tex].

This means that initially vx should be equal to vy, to cover a distance with the lowest energy.

The kinetic energy of the object becomes

[tex]KE= 0.5 m \cdot D\cdot g [/tex].

According to the work-energy theorem, so much work had to be done on it. Work is force times distance along it acts, 2 m in our case, so

[tex]F = 0.25*m*D*g [/tex] .

ehild
 
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  • #11
ehild. John is grade 11. calculus is not an option in Canada 'til grade 12 (unless you are part of the AP/IB curriculum, in which you do vectors in grade 11 as well.) However, you do prove a good point that can be logically reasoned out without calculus.

"This means that initially vx should be equal to vy, to cover a distance with the lowest energy."

Meaning the throwing angle must be 45 deg. With this angle one can resolve the question quite neatly as you follow in your analysis.
 
  • #12
Well, the 45° angle as most effective to throw something at farthest distance might be mentioned in John class. I derived it for the sake of the other possible readers.

ehild
 
  • #13
ehild, I think you must use latex code because it is very hard to read and understand.
 
  • #14
http://physics.wku.edu/~barzilov/phys250/projectile_motion_picture.jpg

As you can see in the image above that [tex]v_0[/tex] is resolved into two rectangular component [tex]v_x[/tex] and [tex]v_{0y}[/tex] and the trajectory of anybody in this case is parabola.

Now you can see that the acceleration along x-axis is zero and on y-axis the acceleration due to gravity is acting downward.

Now, as you are new to this topic I must tell you what happens with the objects' velocity when you throw it up in the air. The vertical velocity continuously decrease with the rise in height due to acceleration due to gravity and then after some time the velocity become zero and then particle start falling downward.

So when object is at his maximum height the object's vertical velocity is zero.

use equation of motion to find maximum height.

v = 0 [tex] u = v_{0}sin\theta[/tex] and a = -g

[tex] v^2 = u^2 + 2aH[/tex]
[tex] 0 = u^{2}sin^{2}\theta - 2gH[/tex]
[tex] H = \frac{u^{2}sin^{2}\theta}{2g}[/tex]

time taken to reach that maximum height can also be calculated

[tex] v = u + at[/tex]
[tex] 0 = u sin\theta - gt[/tex]
[tex] t = \frac{u sin\theta}{g}[/tex]

as time of ascent is equal to time of descent

Time of flight = Time of ascent + Time of descent

[tex]T = \frac{2u sin\theta}{g}[/tex]

and Range = horizontal velocity multiplied of Time of flight

[tex] R = u cos\theta \frac{2u sin\theta}{g}[/tex]
[tex]R = \frac{u^2 Sin 2\theta}{g}[/tex]
 
Last edited by a moderator:
  • #15
to snshusat161, what does the u mean in your equations?
 

1. What is the work-energy theorem?

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.

2. How do you use the work-energy theorem to solve problems?

To solve problems using the work-energy theorem, you first need to identify the initial and final states of the object, and then calculate the work done on the object by all forces using the equation W = Fd. Next, you can use the equation KE = 1/2mv^2 to calculate the change in kinetic energy. Finally, you can equate the work done to the change in kinetic energy to solve for the unknown variable.

3. What are some examples of problems that can be solved using the work-energy theorem?

Examples of problems that can be solved using the work-energy theorem include calculating the speed of an object after it has been pushed a certain distance, determining the force required to stop a moving object in a given distance, and finding the height of a hill that a skier can reach given their initial speed and the coefficient of friction.

4. How does work relate to energy in the work-energy theorem?

In the work-energy theorem, work is a measure of the energy transferred to or from an object. When work is done on an object, its energy changes. This change in energy is equal to the work done on the object.

5. What are the units of measurement for work and energy in the work-energy theorem?

Work is measured in joules (J), while energy is also measured in joules (J). This is because work and energy are essentially interchangeable in the work-energy theorem - work is the transfer of energy from one form to another.

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