2 cars of equal mass m collide at an intersection. Driver E traveled eastward and driver N northward. After the collision, 2 cars remain joined together and slide, before coming to a rest. Police measured the skid mark length d to be 9metres. Coefficient of friction is 0.9. based on the skid marks find the 2 joined cars speed.
To find the speed of the joined cars after the collision, we can use the principles of conservation of momentum and friction.
1. Conservation of Momentum:
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Momentum = Mass x Velocity (p = mv)
Let's assume the initial velocity of the eastward car (Driver E) as Ve and the initial velocity of the northward car (Driver N) as Vn. Since both cars have equal mass m, their momentum before the collision is given by:
Momentum before collision = (mass x velocity of Driver E) + (mass x velocity of Driver N)
= m * Ve + m * Vn
= m(Ve + Vn)
After the collision, the two cars join together and slide. Let's assume their final velocity as Vf.
Momentum after collision = mass x final velocity of joined cars
= m * Vf
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
m(Ve + Vn) = m * Vf
2. Coefficient of Friction:
The coefficient of friction (μ) can be used to calculate the force of friction (Ff) between the joined cars and the road. The force of friction is given by:
Ff = μ * Normal force
Normal force (Fn) is the force exerted by the road on the cars, which is equal to the gravitational force (mg) acting on each car, where g is the acceleration due to gravity.
3. Skid Mark Length:
The skid mark length (d) can be related to the force of friction and the deceleration (a) of the cars using the equation:
d = ((initial velocity)^2 - (final velocity)^2) / (2 * acceleration)
Rearranging this equation, we can get the deceleration (a) in terms of d:
a = ((initial velocity)^2 - (final velocity)^2) / (2 * d)
Now, we can use these principles to find the initial velocity (Ve + Vn) and the final velocity (Vf) of the joined cars.
First, calculate the deceleration:
a = ((initial velocity)^2 - (final velocity)^2) / (2 * d)
= ((Ve + Vn)^2 - Vf^2) / (2 * d)
The force of friction (Ff) is calculated as:
Ff = μ * Normal force
= μ * (mass * g)
= μmg
The net force acting on the joined cars is equal to the force of friction:
Fnet = Ff
Using Newton's second law of motion, Fnet = mass x acceleration:
mass x acceleration = Ff
m * a = μmg
Now substitute the value of the deceleration (a) in the equation above:
m * ((Ve + Vn)^2 - Vf^2) / (2 * d) = μmg
Rearrange the equation to solve for the final velocity (Vf):
((Ve + Vn)^2 - Vf^2) = 2μgd
Simplify the equation:
(Ve + Vn)^2 - Vf^2 = 2μgd
Expand the equation:
Ve^2 + Vn^2 + 2VeVn - Vf^2 = 2μgd
Now, substitute the value of Ve = Vn from the equation m(Ve + Vn) = m * Vf obtained from the conservation of momentum:
(2Vn)^2 - Vf^2 = 2μgd
Simplify further:
4Vn^2 - Vf^2 = 2μgd
Finally, solve for Vf:
Vf^2 = 4Vn^2 - 2μgd
Vf = sqrt(4Vn^2 - 2μgd)
By substituting the given values of Vn, μ, and d, you can calculate the speed of the joined cars after the collision.