A 100g toy car is propelled by a compressed spring that starts it moving.The compression of the spring is 18cm. It coasts up the frictionless slope, gaining 0.2 min altitude.The speed of the car at this altitude is 0.5 m/s. Find the spring constant.
Would the way to approach this be
.5mv^2 = mgh + .5kx^2
I'm stuck because it doesn't provide max height, so there is still both kinetic energy and potential energy, and I'm not sure if I have put it together correctly
No that is not the way.
first, find the initial KE: it is equal to the 1/2 kx^2
then, that has to equal the energy at .2 altitude.
1/2 kx^2=mgh + 1/2 mv^2
Yes, you are on the right track in using the conservation of mechanical energy to solve this problem. The equation you wrote, .5mv^2 = mgh + .5kx^2, is a correct expression of the conservation of mechanical energy. Let's go through the steps to solve the problem:
1. First, identify the given information:
- Mass of the car, m = 100g = 0.1kg
- Spring compression, x = 18cm = 0.18m
- Final speed of the car, v = 0.5 m/s
- Change in altitude, h = 0.2m
2. Now, let's determine the potential energy at the maximum height. Since the slope is frictionless, all the initial potential energy is converted into kinetic energy at the maximum height. Therefore, mgh in the equation represents the initial potential energy.
3. Next, let's consider the kinetic energy of the car at the maximum height. We are given the speed at this point. So, .5mv^2 represents the kinetic energy.
4. Now, update the equation with the given values:
.5mv^2 = mgh + .5kx^2
0.5 * 0.1kg * (0.5m/s)^2 = 0.1kg * g * 0.2m + 0.5 * k * (0.18m)^2
5. At this point, we need to simplify the equation and solve for k (the spring constant). To do that, find the known values:
- g, the acceleration due to gravity, is approximately 9.8 m/s^2.
6. Now, rearrange the equation to solve for k:
.5 * (0.1kg * (0.5m/s)^2 - 0.1kg * g * 0.2m) = 0.5 * k * (0.18m)^2
Simplify the left side:
0.5 * (0.1kg * (0.25m/s^2 - 0.2m/s^2)) = 0.5 * k * (0.18m)^2
0.5 * (0.1kg * 0.05m/s^2) = 0.5 * k * (0.18m)^2
0.0025 kg * m/s^2 = 0.5 * k * (0.0324 m^2)
7. Now, solve for k:
k = (0.0025 kg * m/s^2) / (0.5 * (0.0324 m^2))
k ≈ 0.1529 N/m
Therefore, the spring constant is approximately 0.1529 N/m.