60 Linear Momentum and Force

Example 1. Calculating Momentum: A Football Player and a Football

1. Calculate the momentum of a 110-kg football player running at 8.00 m/s.
2. Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.

Strategy

No information is given regarding direction, and so we can calculate only the magnitude of the momentum, p. (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes p = mv when only magnitudes are considered.

Solution for Part 1

To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation.

p_player = (110 kg)( 8.00 m/s) = 880 kg · m/s

Solution for Part 2

To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation.

p _ball = (0.410 kg)(25.0 m/s) = 10.3 kg · m/s

The ratio of the player’s momentum to that of the ball is

[latex]\displaystyle\frac{p_{\text{player}}}{p_{\text{ball}}}=\frac{880}{10.3}=85.9\\[/latex]

Discussion

Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.

Example 2. Calculating Force: Venus Williams’ Racquet

During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?

Strategy

This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as

[latex]\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}[/latex]

As noted above, when mass is constant, the change in momentum is given by Δ p = mΔv = m ( v _f − v _i ) .

In this example, the velocity just after impact and the change in time are given; thus, once Δp is calculated, [latex]{\mathbf{F}}_{\text{net}}=\frac{\Delta{p}}{\Delta t}[/latex] can be used to find the force.

Solution

To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.

[latex]\begin{array}{lll}\Delta{p}&=&m(v_{\text{f}}-v{\text{i}})\\ &=&(0.057\text{ kg})(58\text{ m/s}-0\text{ m/s})\\ &=&3.306\text{ kg}\cdot\text{m/s}\approx3.3\text{ kg}\cdot\text{m/s}\end{array}\\[/latex]

Now the magnitude of the net external force can determined by using [latex]{\mathbf{F}}_{\text{net}}=\frac{\Delta{p}}{\Delta t}[/latex]:

[latex]\begin{array}{lll}\mathbf{F}_{\text{net}}&=&\frac{\Delta{p}}{\Delta{t}}=\frac{3.306\text{ kg}\cdot\text{m/s}}{5.0\times10^{-3}\text{ s}}\\ &=&661\text{ N}\approx660\text{ N}\end{array}\\[/latex]

where we have retained only two significant figures in the final step.

Discussion

This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using F_net = ma, but one additional step would be required compared with the strategy used in this example.