A 2000 kg is released from a height 20m, and hits a 700kg car at the bottom of the swing. The car is sent horizontally 30, before coming to a stop, and the ball is going 12m/s immediately after they hit. What force of friction is acting between the car and the ground?
To find the force of friction acting between the car and the ground, we need to analyze the motion of the system when the ball and the car collide.
First, let's calculate the potential energy (PE) of the ball at the top of the swing and convert it to kinetic energy (KE) when it reaches the bottom:
PE = mgh
PE = 2000 kg × 9.8 m/s^2 × 20 m
PE = 392,000 J
The PE of the ball is then converted into the KE of the ball:
KE = 392,000 J
Now, let's determine the velocity of the ball just after the collision. Since we know the mass of the ball (2000 kg) and its velocity (12 m/s), we can use the conservation of momentum to find the velocity of the combined system (ball + car) after the collision.
Let v_b be the velocity of the ball after the collision and v_c be the velocity of the car after the collision.
Conservation of momentum in the horizontal direction states:
m_b × v_b + m_c × v_c = m_b × v_bn + m_c × 0
(2000 kg) × (12 m/s) + (700 kg) × v_c = (2000 kg + 700 kg) × v1
24,000 kg·m/s + 700 kg × v_c = 27,000 kg × v_c
27,000 kg × v_c - 700 kg × v_c = 24,000 kg·m/s
26,300 kg × v_c - 27,300 kg × v_c = -3,000 kg·m/s
v_c = -3,000 kg·m/s / (-1,300 kg)
v_c ≈ 2.31 m/s
After the collision, the car moves horizontally at a velocity of approximately 2.31 m/s.
Now, let's calculate the change in kinetic energy (ΔKE_car) of the car from the moment of collision until it comes to a stop. The change in kinetic energy can be expressed as:
ΔKE_car = KE_final - KE_initial
ΔKE_car = 0 J - (1/2) × m_c × v_c^2
ΔKE_car = - (1/2) × (700 kg) × (2.31 m/s)^2
ΔKE_car ≈ -1918.375 J
The change in kinetic energy (ΔKE_car) is also equal to the work done by the friction force (W_friction) acting on the car as it comes to a stop. Since work can be calculated by the equation:
W_friction = force_friction × distance
We can rearrange the equation to find the force of friction (force_friction):
force_friction = ΔKE_car / distance
Given that the car moves horizontally for a distance of 30 m, we can now calculate the force of friction:
force_friction = (-1918.375 J) / (30 m)
force_friction ≈ -63.95 N
The force of friction acting between the car and the ground is approximately 63.95 Newtons, directed opposite to the car's motion.