#### Inclined plane

A box is released from rest at the top of a 30 degree ramp of length 3 m. The box slides down the ramp, dropping a vertical distance of 1.5 m to the floor. Neglect friction. How long does it take the box to reach the floor?

Use DID TASC
to solve for acceleration, then kinematic equations to solve for everything else.
1. Diagram and coordinate system
2. Isolate the system
3. Draw all forces acting
4. Take components
5. Apply F=ma
6. Solve
7. Check!

Steps 1-3: See above diagram

Step 4: Take components
 max = mg sin(θ) may = mg cos(θ)

Step 5: Apply Newton's 2nd law

ΣFx = m ax gives mg sin(θ) = m ax

Step 6: Solve

ax = g sin(θ)

Plugging in g = 9.8 and θ = 30 degrees gives: ax = 4.9 m/s2

Step 7: Check!   &theta=0; gives ax=0; &theta=90o gives ax=g.

Finally: How long does it take the box to reach the floor?

Plug ax = 4.9 m/s2, x(0) = 0, vx(0) = 0, and x = 3 m, into the kinematic equation:

x(t) = x(0) + vx(0) t + ½ ax t2

This gives 3 = 2.45 t2

Solving for t gives 1.11 s as the time for the box to slide down the ramp.