AP PHYSICS: Force and Deceleration

Q: A vehicle weighing ( F_w ) 17.08 kN moves at a constant velocity ( v ) of 35.8 m/s. At some point, the driver decides to let the vehicle coast in neutral, during which air drag causes it to decelerate to 22.4 m/s in 24 s. ( a ) What is the magnitude of deceleration during the period? ( b ) Determine the magnitude of the air drag resisting the car’s forward motion?

A: We begin with a brief analysis of the units of each variable in the all-too-familiar force equation:

F = ma

The mass ( m ) term is expressed in kilograms ( kg ), and acceleration ( a ) is a measure of the degree to which an object’s velocity ( v ) increases or decreases each second ( s ). Furthermore, velocity is measured in meters per second ( m / s ). Let’s now use this information to rewrite the force equation expressed as an average force ( F_avg ) rewritten in terms of momentum ( p ).

F_avg = ( m )( a_avg )

a_avg = ( Δv / Δt )

F_avg = ( m )( Δv / Δt )

Δp = mΔv

F_avg = ( Δp / Δt )

Please recall that kinetic energy ( KE ) can be rewritten in terms of momentum:

KE = ½ mv²

p² = m²v²

½ p² = ½ m²v²

KE = ½ mv² = ( p² / 2m )

Thus, if an unbalanced force causes the momentum of an object to change, it by default causes a change in its kinetic energy as well. These considerations are of general interest to us, but this information isn’t implicitly needed to answer the question at all. All we need to do is determine how much the object’s velocity changed and divide this value by the time over which the deceleration occurred:

Δp = p_f – p_i

mΔv = mΔv_f – mΔv_i

Δv = Δv_f – Δv_i

Δv = 22.4 m/s – 35.8 m/s

Δv = – 13.4 m/s

a_avg = ( Δv / Δt )

a_avg = [ ( – 13.4 m/s ) / 24 s ]

a_avg = – 0.56 m/s²

In order to determine the drag force acting upon the object, we need to know its mass. We can find the mass of the object with relative ease if we realize that its weight ( F_w ) is a product of its mass and the gravitational constant of acceleration ( g ) of an object close to the earth’s surface:

F_w = mg

m = ( F_w / g )

m = ( 17.08 kN / 9.86 m/s² )

m = 1,742.9 kg

F_avg = ( m )( a_avg )

F_avg = ( 1,742.9 kg )( – 0.56 m/s²  )

F_avg = – 976.0 N