In a classic laboratory experiment, a cart of mass m1 on a horizontal air track is attached via a string over an ideal pulley to a hanging cylinder of mass m2. the system is released from rest, and the motion is measured. find the speed of the cart after the cylinder has descended a distance h. (use the following as necessary: m1, m2, h, and g.)

Explanation:

Newton's second law:

∑F = m*a Formula (1)

∑F : algebraic sum of the forces in Newton (N)

m : mass s (kg)

a : acceleration  (m/s²)

We define the x-axis in the direction parallel to the movement of the cart and the y-axis in the direction the movement of the cylinder

Forces acting on the cart

W₁: Weight of the cart : In vertical direction  (-y)

N₁ : Normal force :In vertical direction  (+y)

T: tension force:  In horizontal direction (+x)

Newton's second law  to the cart:

∑Fx = m₁*a

T = m₁*a Equation(1)

Forces acting on the cylinder

W₂= m₂*g : Weight of m₂ : In vertical direction  

T : Tension force: In vertical direction

Newton's second law  to the cylinder :

We take as positive the forces that go in the same direction of the movement of  the cylinder, that is, downaward:

∑Fy =  m₂*a

W₂-T =  m₂*a

W₂- m₂*a  = T Equation(2)

Equation(1)= Equation(2) =T

m₁*a=W₂- m₂*a

m₁*a+m₂*a = W₂

a( m₁+m₂) = m₂*g

Speed of the cart after the cylinder has descended a distance H

We apply the kinematic equation to the cart:

vf²=v₀²+2*a*d formula (2)

Where:  

d:displacement in meters (m)  

v₀: initial speed in m/s  

vf: final speed in m/s  

a: acceleration in m/s²

Data

v₀=0

d = H

We replace data in the formula (2)

vf₁²=v₀₁²+2*a*H

answer and explanation;

air exerts forces on falling objects near earth’s surface.

explanation:

the term 'free falls' refer to object that are falling towards the earth, in a situation in which gravity is the only force acting on the object.

for an object in free fall, the acceleration is constant (g, the gravitational acceleration), and so the velocity of the object constantly increases.

for objects near earth's surface, there is always another force acting on the object: the resistance due to the air. this force is opposed to the direction of fall of the object, and it is proportional to the velocity of the object, so as the object falls down, this force increases up to a point when it balances the force of gravity, and the object continues its motion at constant velocity (called terminal velocity). since in this case gravity is not the only force acting on the object, we are not in a situation of free fall.