can someone please provide explanation? An 800 kg car can roll up and down a ram

Question

can someone please provide explanation?

An 800 kg car can roll up and down a ramp at an angle of theta = 45degree to the ground. It is connected, by a massless towrope that runs over a frictionless pulley, to a heavy spring of spring constant k = 50 kN/m, as shown in the figure. The car handbrake is released when the car is at rest, with the spring in its un-stretched state. Assume the rolling motion of the car is frictionless. What is the speed of the car when it has moved 10 cm down the ramp? How far down the ramp from its point of release does the car roll before momentarily stopping? What is the magnitude of the car's acceleration at the instant the car momentarily stops? What is the direction (up or down the incline) of the car's acceleration at the instant the car momentarily stops?

Explanation / Answer

a) change in potential energy = change in kinetic energy

if the car moves 10 cm along the incline, the its center of mass has moved 10sin(45) cm along the y-direction

=> m*g*10*sin(45) = 0.5*m*v^2

=> v = 1.177 m/s

b) Equate Tension = Spring force to the component of Gravitational force along the incline, we get

k*x = m*g*sin(45)

x = 11.08 cm

c) car's acceleration = g*sin(45) = 6.93 m/s^2

d) acceleration is upwards

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