Impulse, Momentum, Accel
Consider a 0.75 kg wagon (Wagon 1) is pulled forward with a 2 N force for 2 seconds. Consider a second 0.75 kg wagon (Wagon 2) is pulled forward with a 4 N force for 1 second.
- Does Wagon 1 or Wagon 2 have the greater acceleration?
- Does Wagon 1 or Wagon 2 have the greater impulse?
- Does Wagon 1 or wagon 2 have the greater momentum change?
Expand Hint
$$$Force=mass\cdot acceleration=mass\cdot \frac{\Delta velocity}{\Delta time}$$$
Hint 2
Impulse is defined as
$$force\cdot time$$
, while momentum is defined as
$$mass\cdot velocity$$
, and the change in momentum is
$$mass\cdot \Delta velocity$$
.
Newton's second law:
$$$Force=mass\cdot acceleration=mass\cdot \frac{\Delta velocity}{\Delta time}$$$
We can rearrange the above equation to solve the acceleration for both wagons:
$$$a_1=\frac{F_1}{m_1}=\frac{2N}{0.75kg}=2.67\:m/s^2$$$
$$$a_2=\frac{F_2}{m_2}=\frac{4N}{0.75kg}=5.33\:m/s^2$$$
(1.) Wagon 2 has the greater acceleration.
Impulse is defined as
$$force\cdot time$$
, while momentum is defined as
$$mass\cdot velocity$$
, and the change in momentum is
$$mass\cdot \Delta velocity$$
. Therefore, if we multiple both sides of the second law equation by
$$\Delta t$$
:
$$$Force \cdot \Delta time=mass\cdot \Delta velocity$$$
$$$Impulse=Change\:in\:momentum$$$
$$$Impulse_1=F_1\cdot \Delta t_1=2N\cdot 2s=4\:N\cdot s$$$
$$$Impulse_2=F_2\cdot \Delta t_2=4N\cdot 1s=4\:N\cdot s$$$
(2.) The impulse is the same for both wagons.
(3.) Because momentum change is impulse, both wagons have the same momentum change.
(1.) Wagon 2 has the greater acceleration.
(2.) The impulse is the same for both wagons.
(3.) The momentum change is the same for both wagons.
Time Analysis
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