Two motor vehicles A and B are moving along a horizontal straight road in opposite directions. Vehicle A of mass 3 500 kg is moving with a speed of 14 m/s and vehicle B, mass 2 000 kg is moving with a speed 20 m/s. They collide and move together. [4] Calculate the change in kinetic energy.
To calculate the change in kinetic energy, we need to find the initial kinetic energy and the final kinetic energy.
The initial kinetic energy of vehicle A is given by the equation:
K.E. = 0.5 * mass * speed^2
Substituting the values:
K.E. = 0.5 * 3500 kg * (14 m/s)^2 = 686,000 J
The initial kinetic energy of vehicle B is given by the equation:
K.E. = 0.5 * mass * speed^2
Substituting the values:
K.E. = 0.5 * 2000 kg * (20 m/s)^2 = 400,000 J
After the collision, the vehicles move together with a final velocity. To find the final velocity, we use the conservation of momentum:
(mass A * velocity A) + (mass B * velocity B) = (mass A + mass B) * final velocity
(3500 kg * 14 m/s) + (2000 kg * (-20 m/s)) = (3500 kg + 2000 kg) * final velocity
49,000 kg*m/s + (-40,000 kg*m/s) = 5500 kg * final velocity
9000 kg*m/s = 5500 kg * final velocity
final velocity = 9000 kg*m/s / 5500 kg = 1.636 m/s
The final kinetic energy is given by:
K.E. = 0.5 * (mass A + mass B) * final velocity^2
Substituting the values:
K.E. = 0.5 * (3500 kg + 2000 kg) * (1.636 m/s)^2 = 11,242.5 J
The change in kinetic energy is given by:
Change in K.E. = final kinetic energy - initial kinetic energy
Change in K.E. = 11,242.5 J - 686,000 J = -674,757.5 J.
Therefore, the change in kinetic energy is -674,757.5 J.
To calculate the change in kinetic energy, we need to find the initial and final kinetic energies and then subtract the initial from the final.
Given:
Mass of vehicle A (mA) = 3,500 kg
Velocity of vehicle A (vA) = 14 m/s
Mass of vehicle B (mB) = 2,000 kg
Velocity of vehicle B (vB) = 20 m/s
Step 1: Calculate the initial kinetic energy.
The initial kinetic energy of vehicle A (KEA) is given by the formula:
KEA = (1/2) * mA * vA^2
KEA = (1/2) * 3,500 kg * (14 m/s)^2
KEA = (1/2) * 3,500 kg * 196 m^2/s^2
KEA = 343,000 J
The initial kinetic energy of vehicle B (KEB) is given by the formula:
KEB = (1/2) * mB * vB^2
KEB = (1/2) * 2,000 kg * (20 m/s)^2
KEB = (1/2) * 2,000 kg * 400 m^2/s^2
KEB = 400,000 J
The total initial kinetic energy (KE_initial) is the sum of the initial kinetic energies of both vehicles:
KE_initial = KEA + KEB
KE_initial = 343,000 J + 400,000 J
KE_initial = 743,000 J
Step 2: Calculate the final kinetic energy.
After the collision, the two vehicles move together with a common velocity.
To find this common velocity (v_common), we can use the principle of conservation of linear momentum:
mA * vA + mB * vB = (mA + mB) * v_common
(3,500 kg * 14 m/s) + (2,000 kg * 20 m/s) = (3,500 kg + 2,000 kg) * v_common
49,000 kg m/s + 40,000 kg m/s = 5,500 kg * v_common
89,000 kg m/s = 5,500 kg * v_common
v_common = (89,000 kg m/s) / (5,500 kg)
v_common ≈ 16.18 m/s
The final kinetic energy of the moving masses (KE_final) is given by the formula:
KE_final = (1/2) * (mA + mB) * v_common^2
KE_final = (1/2) * (3,500 kg + 2,000 kg) * (16.18 m/s)^2
KE_final = (1/2) * 5,500 kg * 262.07 m^2/s^2
KE_final ≈ 717,938.25 J
Step 3: Calculate the change in kinetic energy.
The change in kinetic energy (ΔKE) is given by the formula:
ΔKE = KE_final - KE_initial
ΔKE = 717,938.25 J - 743,000 J
ΔKE ≈ -25,061.75 J
Therefore, the change in kinetic energy is approximately -25,061.75 J.