Spring compression as a launcher and velocity at a certain point

[pech0706]

Homework Statement

A game uses a spring launcher to shoot a puck with mass of 0.5kg along a frictionless track as shown in the diagram. The spring has a k value of 600 N/m. By how much does the spring need to be compressed to launch the puck and have it go around the inside of the loop without falling off (ignore any spin of the puck)? What is the velocity of the puck as it reaches point C?

Homework Equations

a=v²/R
v²=gR
1/2mv_o²=1/2mv_B²+mgh

The Attempt at a Solution

I could not figure out how to solve this problem. I got v=2.2m/s using the second equation.
Thats all the farther i got.

 

[G01]

Homework Statement

A game uses a spring launcher to shoot a puck with mass of 0.5kg along a frictionless track as shown in the diagram. The spring has a k value of 600 N/m. By how much does the spring need to be compressed to launch the puck and have it go around the inside of the loop without falling off (ignore any spin of the puck)? What is the velocity of the puck as it reaches point C?

Homework Equations

a=v²/R
v²=gR
1/2mv_o²=1/2mv_B²+mgh

The Attempt at a Solution

I could not figure out how to solve this problem. I got v=2.2m/s using the second equation.
Thats all the farther i got.

OK. So, you have the initial speed using conservation of energy. (I'm not sure if the numerical result is correct, but if you show your calculation, I'll check.)

Now use conservation of energy again, to find the initial potential energy in the spring. You should be able to get to the answer from there.

 

[pech0706]

i used v²=gR to find the v at B. then i used that in the energy conservation equation and got v_o to be 5.85. and i know that the V_o=V_c, so do those numbers look right?

 

[G01]

All looks good to me.

Now, like I said, using conservation of energy you should be able to find the compression of the spring using the information you now have.

 

[rwisz]

I would approach this problem by first identifying the relevant equations and variables. From the given information, we can use the equations for centripetal acceleration (a=v^2/R) and gravitational potential energy (PE=mgh) to find the velocity and height of the puck at point C. We can also use the conservation of energy equation (KEi + PEi = KEf + PEf) to find the initial velocity of the puck at point A.

To find the required spring compression, we can use the equation for potential energy stored in a spring (PEs=1/2kx^2) and set it equal to the potential energy of the puck at point A (PE=mgh). Solving for x, we can find the amount of compression needed for the spring.

Using the given mass and k value, we can then plug in the values for x and solve for the velocity of the puck at point C using the equation v=sqrt(2gh). This will give us the velocity of the puck at the top of the loop.

However, to ensure that the puck does not fall off the loop, we also need to consider the minimum velocity required for circular motion at the top of the loop. This can be found using the equation v=sqrt(gR). If the velocity at point C is greater than this minimum velocity, then the puck will remain on the loop.

In conclusion, by using the relevant equations and considering the conditions for circular motion, we can determine the required spring compression and velocity of the puck at point C in order for it to successfully go around the loop without falling off.