Question

A truck of mass M and a bike of mass m approach a stop sign on a rainy day. As they approach the stopsign, both vehicles are travelling with speed vo. When their front tires are even with the stopsign, the drivers each engage their brakes as start to skid. Each tire has a coefficient of kinetic friction mu_k with the ground. Given M, m, vo, g, how far does each object slide into the intersection? In your skeptic section, pick reasonable values for M, m, mu_k (you can use g = 9.8 m/s^2) and interpret your answers. Is your answer reasonable?

Answer 1

Since both bike and truck are moving horizontally, they have no vertical acceleration and hence by Newton's second law, vertical forces must be balanced .

Vertical forces on bike includes:

1.gravitational force, mg downwards , where m is mass of bike, g is gravitational acceleration.

2. Normal reaction upwards

So, normal reaction on bike=mg

Similarly, normal reaction on truck=Mg as mass of truck is M, g is gravitational acceleration.

Frictional force=_kN, where _k is coefficient of friction, N is normal reaction.

So, frictional force on bike= _kmg

Frictional force on truck=_kMg

Acceleration=force/mass

So, acceleration of bike=_kmg/m=_kg

Similarly, acceleration of truck=_kMg/M=_kg

So, acceleration of bike is same as that of truck.

Now, friction acts against relative motion. So, direction of friction is opposite to direction of velocity of bike/truck.

Under uniform acceleration,v²=u²+2as, where v is final velocity, u is initial velocity,a is acceleration,s is displacement

Here,for both bike and truck, v=0 m/s as finally vehicles stop, u=v₀ , a=-_kg (negative sign denotes acceleration is opposite to initial velocity u)

So, 0=v₀²-2_kgs=>s=v₀²/(2_kg) = v₀²/(19.6_k) as g=9.8 m/s²

Interpretation: Distance travelled before coming to a stop is same for both bike and truck irrespective of their masses. This is because despite having different masses, they have same acceleration as explained earlier.Also, they have same initial velocities and final velocities.So, distance travelled before coming to stop is same for both bike and truck. Hence, the answer is reasonable.