A baseball of mass m1 = 0.46 kg is thrown at a concrete block m2 = 8.25 kg. The block has a coefficient of static friction of μs = 0.81 between it and the floor. The ball is in contact with the block for t = 0.165 s while it collides elastically.
m1 = 0.46 kg
m2 = 8.25 kg
μs = 0.81
t = 0.165 s
Part (a) Write an expression for the minimum velocity the ball must have, vmin, to make the block move.
Part (b) What is the velocity in m/s?
Part (a) To determine the minimum velocity of the ball required to make the block move, we need to consider the frictional force acting on the block. The maximum frictional force, known as static friction, can be calculated using the equation:
Fstatic = μs * N
where Fstatic is the maximum static frictional force, μs is the coefficient of static friction between the block and the floor, and N is the normal force acting on the block (equal to its weight).
Since the ball is in contact with the block during the collision, it exerts a normal force on the block. The normal force can be calculated as:
N = m2 * g
where m2 is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²).
Now, during the collision, the ball will stick to the block momentarily and impart its momentum to the block. The momentum of the system before the collision is given by:
p_initial = m1 * v_initial
where m1 is the mass of the ball and v_initial is its initial velocity.
To set the block in motion, the momentum of the system after the collision must be equal to or greater than the maximum static frictional force. Since the ball and block move with the same velocity after the collision (since they stick together), the momentum of the system after the collision is given by:
p_final = (m1 + m2) * v_final
where m1 + m2 is the combined mass of the ball and block and v_final is their final velocity after the collision.
Therefore, to make the block move, the initial momentum must be equal to or greater than the final momentum:
m1 * v_initial ≥ (m1 + m2) * v_final
Now, we can solve for the minimum initial velocity (v_initial) required to make the block move:
v_initial ≥ (m1 + m2) * v_final / m1
However, since the collision is elastic, the final velocity of the system can be determined using the conservation of momentum equation:
m1 * v_initial = (m1 + m2) * v_final
We can rearrange this equation to solve for v_final:
v_final = m1 * v_initial / (m1 + m2)
Therefore, substituting the expression for v_final into the equation for v_initial, we get:
v_initial ≥ (m1 + m2) * (m1 * v_initial) / (m1 + m2) / m1
Simplifying further, we obtain:
v_initial ≥ v_initial
So, the minimum initial velocity required to make the block move is simply any positive value of v_initial.
Part (b) Since the minimum initial velocity required to make the block move can be any positive value, the velocity in m/s is not uniquely determined and can be chosen freely as long as it is greater than zero.
To answer part (a), we need to consider the force required to overcome the static friction between the block and the floor. The force of static friction is given by the equation:
Fs = μs * N
Where Fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force exerted on the block by the floor.
Since the block is at rest, the normal force is equal to its weight:
N = m2 * g
Where g is the acceleration due to gravity.
Let's substitute these equations into the expression for the force:
Fs = μs * N
= μs * m2 * g
Now, let's consider the collision between the ball and the block. Since the collision is elastic, the total momentum before the collision is equal to the total momentum after the collision:
m1 * vb = (m1 + m2) * v'
Where vb is the velocity of the ball before the collision and v' is the velocity of the ball and the block after the collision.
Since we are looking for the minimum velocity, we can assume that the block just begins to move. This means that the force of static friction is equal to the maximum force of static friction:
Fs = μs * m2 * g
And the force of static friction can be calculated as:
Fs = Δp / Δt
Where Δp is the change in momentum and Δt is the time of collision.
Now, we can solve for the minimum velocity, vmin:
Fs = Δp / Δt
μs * m2 * g = (m1 + m2) * vmin
Therefore, the expression for the minimum velocity is:
vmin = (μs * m2 * g) / (m1 + m2)
To answer part (b), we can substitute the given values into the expression for vmin:
vmin = (0.81 * 8.25 * 9.8) / (0.46 + 8.25)
Calculating this expression will give us the velocity in m/s.