Final answer:
The time it takes to stop a 0.20 kg baseball moving at 20.60 m/s with a force of -350 N is approximately 0.0118 seconds. To calculate the distance the baseball travels before stopping, we calculate the change in kinetic energy and find the distance to be approximately 12.07 cm.
Step-by-step explanation:
Calculating Time to Stop a Baseball
A 0.20 kg baseball moving at 20.60 m/s is slowed to a stop by a catcher who exerts a constant force of -350 N. To calculate how long it takes for the force to stop the ball, we can use the impulse-momentum theorem. The impulse-momentum theorem states that the change in momentum (Δp) of an object is equal to the impulse (J) applied to it, where impulse is the product of the average force (F) applied and the time duration (Δt) over which it is applied.
Impulse-Momentum Theorem Equation: J = Δp = F Δt
To find the time (Δt), we rearrange the equation to solve for Δt: Δt = Δp / F. The change in momentum (Δp) is the final momentum (p_f) minus the initial momentum (p_i). Since the ball comes to a stop, its final momentum is zero, and the initial momentum is the product of its mass (m) and initial velocity (v_i).
Calculating the Initial Momentum
p_i = m v_i = 0.20 kg × 20.60 m/s = 4.12 kg·m/s
Since the catcher applies a constant force of -350 N in the opposite direction of the ball's motion to bring it to a stop, we get:
Δt = Δp / F = 4.12 kg·m/s / (-350 N) = -0.0118 s (The negative sign indicates that the force is in the opposite direction to the ball's motion, but duration of time is always positive, so we take the absolute value.)
The time it takes to stop the ball is therefore approximately 0.0118 s.
Calculating Distance Traveled Before Stopping
To calculate the distance the ball travels before stopping, we can use the work-energy principle that states work done by all forces acting on an object is equal to the change in its kinetic energy (ΔKE).
Work done (W) by the catcher's force can be found by the formula W = F d, where d is the distance over which the force is applied. The kinetic energy of the baseball when it is moving can be calculated using the formula KE = ½ m v^2. Upon stopping, the kinetic energy is zero, so the change in kinetic energy is simply the negative of the initial kinetic energy.
Calculating the Initial Kinetic Energy
ΔKE = KE_final - KE_initial = 0 - ½ m v^2 = -½ × 0.20 kg × (20.60 m/s)^2 = -42.244 J
Since work done (W) is also the negative change in kinetic energy, we have W = -ΔKE. Thus:
d = W / F = -ΔKE / F = 42.244 J / 350 N = 0.1207 m or 12.07 cm.
The ball travels approximately 12.07 cm before stopping.