A mouse of mass 5.0 g spots the corner of a peanut butter sandwich of mass 8.0 g left on an ice rink after a game. Excited, the mouse runs out onto the ice, but immediately begins to slide. The mouse reaches the peanut butter sandwich and sinks its teeth in. Both the mouse and peanut butter sandwich continue to slide with a speed of 0.45 m/s. What was the initial speed of the mouse?
They want you to neglect friction.
Total momentum is conserved, and equals the initial mouse momentum.
0.005*V0 = (0.005 + 0.008)*0.45
Vo = 13/5 * 0.45 = 1.17 m/s
1.17 m/s
To determine the initial speed of the mouse, we can use the principle of conservation of momentum. The total momentum before the mouse reaches the sandwich is equal to the total momentum after they start sliding together.
The momentum of an object is calculated by multiplying its mass by its velocity. Let's denote the initial velocity of the mouse as "v" and the mass of the mouse as "m1" (5.0 g or 0.005 kg), and the mass of the peanut butter sandwich as "m2" (8.0 g or 0.008 kg).
Before the mouse reaches the sandwich, the momentum of the mouse is given by:
momentum1 = m1 * v
After the mouse reaches the sandwich, both the mouse and the sandwich slide together with a speed of 0.45 m/s. So the momentum of the mouse and the sandwich combined is:
momentum2 = (m1 + m2) * 0.45
According to the conservation of momentum principle, momentum1 = momentum2:
m1 * v = (m1 + m2) * 0.45
Substituting the given values:
0.005 * v = (0.005 + 0.008) * 0.45
0.005v = 0.0135
v = 0.0135 / 0.005
v ≈ 2.7 m/s
Therefore, the initial speed of the mouse was approximately 2.7 m/s.
To find the initial speed of the mouse, we can make use of the principle of conservation of momentum.
The equation for momentum is given by:
p = m * v
Where:
p is the momentum
m is the mass
v is the velocity
According to the problem, the only external force acting on the mouse and the sandwich is the friction force on the ice rink. Therefore, the momentum of the system is conserved. Initially, the mouse is at rest, so its momentum is zero.
After the mouse reaches the sandwich and both start sliding, their final momentum is given by:
p_final = m_mouse * v_final + m_sandwich * v_final
We are given the mass of the mouse (5.0 g) and the mass of the sandwich (8.0 g), both in grams. To use consistent units, we need to convert the masses to kilograms.
m_mouse = 5.0 g = 0.005 kg
m_sandwich = 8.0 g = 0.008 kg
We are also given the final speed v_final = 0.45 m/s.
Using conservation of momentum, we have:
0 = 0.005 kg * v_initial + 0.008 kg * 0.45 m/s
Now, we can solve for v_initial:
v_initial = -(0.008 kg * 0.45 m/s) / 0.005 kg
Calculating this, we get:
v_initial = -0.072 m/s
Therefore, the initial speed of the mouse is -0.072 m/s. The negative sign indicates that the mouse was initially moving in the opposite direction to its final motion.