A force that averages \(984 \mathrm{~N}\) is applied to a \(420-\mathrm{g}\) steel ball moving at \(13.8 \mathrm{~m} / \mathrm{s}\) by a collision lasting \(27.0 \mathrm{~ms}\). If the force is in a direction opposite to the initial velocity of the ball, find the final speed of the ball.

Short Answer

Expert verified
The final speed of the ball is \(77.1 \,m/s\).

Step by step solution

01

Convert quantities to correct units

First, convert the mass of the ball from grams to kilograms by dividing by 1000, so mass \(m = 0.420 \, kg\). Secondly, convert the collision time from milliseconds to seconds by dividing by 1000, thus the time \(t = 0.027 \, seconds.\)
02

Calculate acceleration

Use Newton's second law of motion to find acceleration. The law is \(F = ma\), where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. Rearrange the formula to solve for \(a = \frac{F}{m} = \frac{984N}{0.420kg} = 2343 \, m/s^2\). The negative sign indicates that the acceleration is in the opposite direction to the initial velocity.
03

Calculate change in velocity

Next, calculate the change in velocity due to this acceleration. Use the formula \(\Delta v = a \times t = -2343 \,m/s^2 \times 0.027 \,s = -63.3 \,m/s\). The negative sign indicates the velocity change is opposite to initial velocity.
04

Calculate final speed

Subtract this change in velocity from the initial speed to find the final speed. An important detail here is that speed is a scalar, so it's always positive. Calculate the final speed as \(final \ speed = initial \ speed - \Delta v = 13.8 \,m/s - (-63.3 \,m/s) = 77.1 \,m/s\).

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