Question

each group of students will choose a car and a spring to push the car and then build a track. the assignment is to make the car go 5.0 m/s when it reaches the bottom of the first hill. four groups of students choose springs and build tracks as described in the table.

group	car mass (kg)	spring constant (n/m)	hill height (m)
b	0.60 kg	35 n/m	0.90 m
c	0.55 kg	40 n/m	1.1 m
d	0.84 kg	32 n/m	0.95 m

which groups roller coaster will most likely make the car travel closest to 5.0 m/s when it is at the bottom of the first hill?
a
b
c
d

Explanation:

Step1: Use conservation of energy

The initial energy of the system is the elastic - potential energy of the spring $U_{s}=\frac{1}{2}kx^{2}$ (assuming the spring is compressed and then released) and the gravitational - potential energy $U_{g}=mgh$. The final energy is the kinetic energy $K = \frac{1}{2}mv^{2}$. If we assume the spring is compressed and then released to start the car, and neglecting friction, $\frac{1}{2}kx^{2}+mgh=\frac{1}{2}mv^{2}$. Since we are not given information about the spring - compression $x$, if we assume the spring is just used to start the car and its contribution is relatively small compared to the gravitational - potential energy change, we can mainly consider the conversion of gravitational - potential energy to kinetic energy $mgh=\frac{1}{2}mv^{2}$, so $v=\sqrt{2gh}$.

Step2: Calculate the theoretical velocity for each group

For group A: $v_{A}=\sqrt{2\times9.8\times1.2}\approx4.85\ m/s$.
For group B: $v_{B}=\sqrt{2\times9.8\times0.90}\approx4.20\ m/s$.
For group C: $v_{C}=\sqrt{2\times9.8\times1.1}\approx4.64\ m/s$.
For group D: $v_{D}=\sqrt{2\times9.8\times0.95}\approx4.32\ m/s$.
The velocity of group C is closest to $5.0\ m/s$.

Answer:

C. Option C