in a baseball game a 0.2 kg ball moving at 12 m/s is hit by a bat. after the impact the ball moves in the opposite direction with a velocity of 18 m/s. if the force exerted on the ball of the batter is 670 N. how long is the time of contact
To find the time of contact between the ball and the bat, we can use the impulse-momentum principle.
Impulse is defined as the force applied to an object multiplied by the time it is applied. In this case, the impulse exerted on the ball by the bat is equal to the change in momentum of the ball.
Step 1: Calculate the initial momentum of the ball.
The initial momentum of the ball is given by the formula:
Momentum = mass × velocity
Given:
Mass of the ball (m) = 0.2 kg
Initial velocity of the ball (u) = 12 m/s
Initial momentum (p) = m × u
Step 2: Calculate the final momentum of the ball.
The final momentum of the ball is given by the formula:
Momentum = mass × velocity
Given:
Final velocity of the ball (v) = -18 m/s (opposite direction)
Final momentum (p') = m × v
Step 3: Calculate the change in momentum.
Change in momentum (Δp) = p' - p
Step 4: Calculate the impulse exerted on the ball.
Impulse (J) = Δp
Given:
Force exerted on the ball (F) = 670 N
From the impulse-momentum relationship, we know that:
Impulse = Force × time
J = F × t
Step 5: Solve for the time of contact.
Substitute the given values into the equation and solve for t:
t = J / F
Now let's calculate:
Step 1:
Initial momentum (p) = m × u = 0.2 kg × 12 m/s = 2.4 kg·m/s
Step 2:
Final momentum (p') = m × v = 0.2 kg × -18 m/s = -3.6 kg·m/s
Step 3:
Change in momentum (Δp) = p' - p = -3.6 kg·m/s - 2.4 kg·m/s = -6 kg·m/s
Step 4:
Impulse (J) = Δp = -6 kg·m/s
Step 5:
Time of contact (t) = J / F = -6 kg·m/s / 670 N
Thus, the time of contact between the ball and the bat is approximately t = -0.00896 s.
To find the time of contact, we can use the principles of Newton's second law.
1. First, we need to calculate the change in momentum of the ball. The momentum of an object can be calculated by multiplying its mass by its velocity.
Initial momentum = mass x initial velocity
Final momentum = mass x final velocity
Given that the mass of the ball is 0.2 kg, the initial velocity is 12 m/s, and the final velocity is -18 m/s (opposite direction), we can substitute these values into the equations:
Initial momentum = 0.2 kg x 12 m/s = 2.4 kg⋅m/s
Final momentum = 0.2 kg x (-18 m/s) = -3.6 kg⋅m/s (considering opposite direction)
The change in momentum is calculated by subtracting the initial momentum from the final momentum:
Change in momentum = Final momentum - Initial momentum
Change in momentum = -3.6 kg⋅m/s - 2.4 kg⋅m/s = -6 kg⋅m/s
2. Next, we can use the formula for impulse, which is the force exerted on an object multiplied by the time of contact:
Impulse = Force x Time
Given that the force exerted on the ball is 670 N, we can rearrange the equation to solve for time:
Time = Impulse / Force
Substituting the values:
Time = (-6 kg⋅m/s) / 670 N
Time ≈ -0.009 seconds (approximately)
Therefore, the time of contact is approximately -0.009 seconds. Note that the negative sign indicates a direction opposite to the initial velocity.
physics
Force x (contact time) = Momentum change
= 0.2 kg*(12 + 18)m/s = 6.0 kg*m/s
Contact time = 6.0/670 = 9.0*10^-3 s