Bob, who has a mass of 75 kg, can throw a 500 g rock with a speed of 30 m/s. The
distance through which his hand moves as he accelerates the rock forward from rest
until he releases it is 1.0 m.
a. What constant force must Bob exert on the rock to throw it with this speed?
b. If Bob is standing on frictionless ice, what is his recoil speed after releasing the rock
To answer part (a) of the question, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a). In this case, we need to find the force required to give the rock a certain speed.
1. Start by converting the mass of the rock to kilograms: 500 g = 0.5 kg.
2. Determine the acceleration of the rock. Using the kinematic equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we know that the rock starts from rest (u = 0 m/s) and reaches a final velocity of 30 m/s. The time it takes for this acceleration to occur is not given, but we can use the formula v² = u² + 2as, where s is the distance traveled.
3. Rearrange the formula v² = u² + 2as to solve for acceleration (a):
a = (v² - u²) / (2s)
= (30² - 0²) / (2 * 1)
= 900 / 2
= 450 m/s²
4. Now, we can calculate the force using Newton's second law:
F = m * a
= 0.5 kg * 450 m/s²
= 225 N
Therefore, Bob must exert a constant force of 225 Newtons on the rock to throw it with a speed of 30 m/s.
To answer part (b) regarding Bob's recoil speed after releasing the rock on frictionless ice, we can apply the principle of conservation of momentum.
1. The initial momentum of the system (Bob and the rock) is zero since both are at rest.
2. After the rock is released with a certain momentum, Bob will experience an equal and opposite momentum in order to conserve momentum.
3. The momentum of the rock can be calculated using the equation p = m * v, where p is momentum, m is mass, and v is velocity:
Momentum of the rock = 0.5 kg * 30 m/s
= 15 kg m/s
4. Since momentum is conserved, Bob's momentum will be equal in magnitude but in the opposite direction:
Momentum of Bob = -15 kg m/s
5. Now, we can calculate Bob's recoil velocity using the equation:
Momentum = mass * velocity
-15 kg m/s = 75 kg * V
Solving for V, we get:
V = -15 kg m/s / 75 kg
= -0.2 m/s
Therefore, Bob's recoil speed after releasing the rock will be 0.2 m/s in the opposite direction.