A 14000 N automobile travels at a speed of 50 km/h northward along a street, and a 7500 N sports car travels at a speed of 62 km/h eastward along an intersecting street.
(a) If neither driver brakes and the cars collide at the intersection and lock bumpers, what will the velocity of the cars be immediately after the collision?
magnitude:
direction ° (counterclockwise from due east)
(b) What percentage of the initial kinetic energy will be lost in the collision?
Use conservation of momentum to get the final velocity and direction.
With the final velocity known for both cars, you can compute the kinetic energy loss, and express it as a percentage of the initial KE.
Be sure to convert km/h speeds to m/s.
The car weights (in N) should be converted to masses (in kg) when computing momentum, by dividing by g, but the g will cancel out anyway.
To answer this question, we need to use the principles of conservation of momentum and conservation of kinetic energy.
(a) Let's start by finding the initial momentum of each car. The momentum of an object is defined as the product of its mass and velocity. However, we are not given the mass of the cars, but that's okay because we only need the ratio of the masses.
First, let's convert the speeds of the cars into meters per second (m/s).
Car A: Speed = 50 km/h = 50 * (1000 m / 3600 s) = 13.89 m/s (northward)
Car B: Speed = 62 km/h = 62 * (1000 m / 3600 s) = 17.22 m/s (eastward)
Now, let's assume the mass of Car A is m1 and the mass of Car B is m2.
The momentum of Car A (p1) = m1 * v1 = m1 * 13.89 m/s (northward)
The momentum of Car B (p2) = m2 * v2 = m2 * 17.22 m/s (eastward)
After the collision, the cars will stick together and move together as a single object. Therefore, their final momentum (pf) will be the sum of their initial momenta.
pf = p1 + p2
To find the velocity of the cars after the collision, we can divide the final momentum by the total mass of the system (m1 + m2).
final velocity (v_final) = (p1 + p2) / (m1 + m2)
Now, let's find the direction of the final velocity. Since Car A was moving north and Car B was moving east, the angle between their initial velocities is 90 degrees (counterclockwise from due east). So to find the direction of their final velocity, we need to calculate the angle between their final velocity and due east.
θ = arctan(v_final_y / v_final_x)
where v_final_y is the final velocity in the northward direction and v_final_x is the final velocity in the eastward direction.
(b) To find the percentage of initial kinetic energy lost in the collision, we need to compare the initial and final kinetic energies.
Kinetic energy (K.E.) = (1/2) * mass * velocity^2
initial K.E. of Car A = (1/2) * m1 * v1^2
initial K.E. of Car B = (1/2) * m2 * v2^2
final K.E. of the system = (1/2) * (m1 + m2) * v_final^2
The percentage of initial kinetic energy lost can be calculated as:
% lost K.E. = ((initial K.E. of Car A + initial K.E. of Car B) - final K.E. of the system) / (initial K.E. of Car A + initial K.E. of Car B) * 100
Please note that to fully answer the question, we need the masses of both cars. Without that information, we can only provide the general equations and explain how to calculate the answers once the masses are known.