Question

A 3 kg block (block A) is released from rest at the top of a 20 m long frictionless ramp that is 3 m high. At the same time, an identical block (block B) is released next to the ramp so that it drops straight down the same 3 m. Find the values for each of the following for the blocks just before they reach ground level. (a) gravitational potential energy Block A____J Block B____J (b) kinetic energy Block a____ J Block b____ J (c) speed Block a____ m/s Block b____ m/s (d) momentum Block a____ kg m/s Block b____ kg m/s

Answer 1

(a)Gravitational Potential Energy

Block A = mgh J = 3*9.8*3 = 88.2 J.

Block B = mgh J = 3*9.8*3 =  88.2 J.

(b) and (c) Kinetic energy and speed

Both blocks A and B will have exactly the same velocity and the same kinetic energy when they reach down to the ground 3 meters below whether Block A goes through the ramp or it just is dropped vertically to the ground!

The acting gravity force on block A is = g*sin theta.

Theta can be calculated as follows:

Inverse of sin (3/20) = 0.15 ; therefore theta = 8.62 degrees

g sin 8.62 = 9.8*0.15 = 1.47 m/s^2

V final = V initial + a*t; V initial = 0.

V final = 1.47 *t

Use the equation of displacement:

20 = 0.5*1.47*t^2

20/(0.5*1.47)=27.2 = t^2

t = 5.21 seconds

V final = 1.47*5.21=7.65 m/s.  

Let us now check the final velocity had we dropped block B directly and vertically to the ground.
V final = g*t
Again use the displacement equation:
3 = 0.5 *9.8 *t^2
3/4.9 =t^2
t = 0.782 seconds ( note that the time for both blocks to reach the ground is different.
V final = 9.8*0.782 =7.65 m/s

The Kinetic energy for both blocks are equal; it is = to 0.5*m*V^2 =0.5*3*7.65^2 = 87.78 Joules

(d) Momentum

The momentum for both blocks = mv = 3*7.65 = 22.95 kg.meter/s. One is directed down wards, and the other directed at an angle = 8.62 degrees from the horizontal. The reason for the difference in direction is due to the way the first block (A) was placed when it was released to go through the ramp. Block A was reoriented at 8.62 degrees at the top of the ramp, while block B was oriented to go vertically towards the ground!