In a demonstration in physics class, a 1.2 kg dynamics cart starts from rest at the top of a ramp. the ramp is 2.4 m above the ground. the cart then rolls down to the bottom of the ramp, where it collides with a stationary 1.4 kg dynamics cart. Assume that an elastic head-on collision occurs. calculate the speed of each cart just after the collision.
To calculate the speed of each cart just after the collision, we need to apply conservation of momentum. The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act upon it.
Step 1: Find the initial potential energy of the first cart at the top of the ramp. The potential energy formula is given by:
Potential Energy (PE) = mass (m) × gravity (g) × height (h)
PE = 1.2 kg × 9.8 m/s^2 × 2.4 m
PE ≈ 28.224 Joules
Step 2: Convert the potential energy to kinetic energy at the bottom of the ramp using the formula:
Potential Energy (PE) = Kinetic Energy (KE)
KE = 1/2 × mass (m) × velocity^2 (v^2)
28.224 J = 1/2 × 1.2 kg × v^2
Step 3: Calculate the velocity of the first cart at the bottom of the ramp:
v^2 = (2 × 28.224 J) / 1.2 kg
v^2 ≈ 47.04 m^2/s^2
v ≈ √47.04
v ≈ 6.86 m/s
Step 4: Apply conservation of momentum to find the final velocities after the collision. Since it is an elastic collision, both momentum and kinetic energy are conserved.
Law of Conservation of Momentum:
m1initial * v1initial + m2initial * v2initial = m1final * v1final + m2final * v2final
Where:
m1initial and m2initial are the initial masses of the carts
v1initial and v2initial are the initial velocities of the carts
m1final and m2final are the final masses of the carts
v1final and v2final are the final velocities of the carts
Step 5: Since the first cart is rolling down the ramp, it has more kinetic energy. In the collision, the first cart transfers some kinetic energy to the second cart. However, they exchange momentum since it is a head-on collision.
So, m1initial = 1.2 kg
m2initial = 1.4 kg
v1initial = 6.86 m/s (calculated earlier)
v2initial = 0 m/s (the second cart is stationary)
Assuming the first cart's final velocity is v1final and the second cart's final velocity is v2final, we can rearrange the momentum conservation equation:
m1initial * v1initial + m2initial * v2initial = m1final * v1final + m2final * v2final
(1.2 kg × 6.86 m/s) + (1.4 kg × 0 m/s) = (1.2 kg × v1final) + (1.4 kg × v2final)
8.232 kg·m/s = 1.2 kg × v1final + 0 kg·m/s
8.232 kg·m/s = 1.2 kg × v1final
Since the second cart was stationary before the collision, its final velocity v2final will be equal in magnitude but opposite in direction to the final velocity v1final of the first cart.
v2final = -v1final
Plugging this into the equation:
8.232 kg·m/s = 1.2 kg × v1final + 1.4 kg × (-v1final)
8.232 kg·m/s = 1.2 kg × v1final - 1.4 kg × v1final
8.232 kg·m/s = (1.2 kg - 1.4 kg) × v1final
8.232 kg·m/s = (-0.2 kg) × v1final
v1final ≈ 41.16 m/s
Step 6: Calculate the final velocity of the second cart:
v2final = -v1final
v2final = -(41.16 m/s)
v2final ≈ -41.16 m/s
Therefore, just after the collision, the speed of the first cart is approximately 41.16 m/s, and the speed of the second cart is approximately -41.16 m/s.