We have to work out the accepted value of a roller coaster cart (taking into account friction) when the mass is modified
Here is the answer I got for 56.71 grams:
0=∆EGP+∆EK+Fd
0=mg(hf-hi )+1/2 m(v^2-u^2 )+Fd
0=0.05671 × 9.8 (0.075-0.786)+1/2× 0.05671 (v^2-0^2 )+(0.2184019181 × 1.005505843)
0=-0.395143938+0.028355v^2+0.2196044047
(0.395143938-0.2196044047)/0.028355=v^2
v^2=6.190778815
v=√6.190778815
v=2.488127572
v=2.488m/s
But here is the answer I got for 106.71 grams:
0=∆EGP+∆EK+Fd
0=mg(hf-hi )+1/2 m(v^2-u^2 )+Fd
0=0.10671 × 9.8 (0.075-0.786)+1/2× 0.10671 (v^2-0^2 )+(0.7732989017 × 1.005505843)
0=-0.743533938+0.053355v^2+0.777556564
(0.743533938-0.777556564)/0.053355=v^2
v^2=-0.6353431559
v=√0.6353431559
v=0.7970841586
v=0.797m/s
As the roller coaster has an incline, we were told to work out 'F' by doing: mass × weight × cos45 × mass × weight × tan45
'd' is the distance travelled
'hf' is the final height and 'hi' is the initial height
Can someone please tell me what I have done wrong in these calculations
Based on the provided calculation, it seems like there might be a mistake in your equations. Specifically, it seems like there might be an error in the calculation of the gravitational potential energy (∆EGP) and the kinetic energy (∆EK). Let's go through the correct approach step by step to find the accepted value of the roller coaster cart.
1. Start by calculating the change in gravitational potential energy (∆EGP) using the formula: ∆EGP = mg(hf - hi)
2. Next, calculate the change in kinetic energy (∆EK) using the formula: ∆EK = 1/2 m(v^2 - u^2), where u is the initial velocity (assumed to be 0 in this case).
3. The work done by friction (Fd) can be calculated by multiplying the force of friction (F) by the distance traveled (d).
4. Set up the equation using the principle of conservation of energy: 0 = ∆EGP + ∆EK + Fd
5. Rearrange the equation to solve for the velocity (v):
v = √((-(∆EGP + Fd) * 2) / m)
Now, let's go through the calculation for the 56.71 grams case:
Given: m = 0.05671 kg, hf = 0.075 m, hi = 0.786 m
1. Calculate ∆EGP: ∆EGP = 0.05671 * 9.8 * (0.075 - 0.786)
2. Calculate Fd: Fd = mass * weight * cos(45) * distance = 0.05671 * 9.8 * cos(45) * distance
3. Rearrange the equation 0 = ∆EGP + ∆EK + Fd to solve for ∆EK:
∆EK = -∆EGP - Fd
4. Calculate ∆EK: ∆EK = -(∆EGP + Fd)
5. Substitute the values into the equation: ∆EK = -(∆EGP + Fd)
6. Calculate the velocity (v): v = √((-(∆EGP + Fd) * 2) / m)
By following these steps correctly, you should be able to find the correct value of the velocity for the roller coaster cart.