Interactive physics simulator
Impulse: Force & Time (J = Ft)
Impulse is the measure of force applied over a time interval. Defined as the product of force and duration (J = Ft), impulse is equal to the change in momentum of a body. Learn how extending contact time cushions impacts or how thruster duration drives acceleration.
Impulse Simulator
Experiment with collision softening (longer contact reduces peak forces) and rocket propulsion (applying constant force over time).
Live Telemetry
- Ball Mass (m)
- 0.5 kg
- Impact Speed (v_in)
- 15 m/s
- Rebound Speed (v_out)
- -15 m/s
- Contact Time (t)
- 0.10 s
- Total Impulse (J = Δp)
- 15.0 N·s
- Peak Impact Force
- 300 N
What is Impulse?
In physics, impulse (denoted by the symbol J) is a measure of the total effect of a force acting over a time interval. When a force is applied to an object, it causes a change in the object's velocity, and therefore, its momentum. Impulse is a vector quantity whose direction matches the net force.
Mathematically, for a constant force F applied over a time interval t:
The standard SI unit of impulse is the Newton-second (N·s).
Impulse-Momentum Theorem
Derived directly from Newton's Second Law (F = ma = m * Δv / t), the theorem states:
J = Δp
- The impulse delivered to an object equals its change in momentum: F * t = m * v_f - m * v_i.
- This explains why both a large force acting briefly (like a golf swing) and a small force acting for a long time (like thrusters) can cause the same change in speed.
Cushioning Physics
Why do airbags and gymnastic safety mats prevent severe injuries?
- In a sudden stop, a person's change in momentum (Δp) is constant.
- By utilizing soft padding, the duration of deceleration (t) is lengthened.
- Since F_avg = J / t, increasing the contact time reduces the average and peak impact forces to safe levels.
Contact Time Comparison
Stopping a 60 kg gymnastic landing from 5 m/s (Impulse J = -300 N·s):
| Surface | Time (t) | Avg Force (F) | Peak Force |
|---|---|---|---|
| Concrete floor | 0.01 s | 30,000 N | Extreme (Injurious) |
| Dense rubber mat | 0.06 s | 5,000 N | Moderate |
| Thick foam pit | 0.30 s | 1,000 N | Very Low (Safe) |
Force-Time Graph
Forces during real-world collisions are rarely constant. They rise to a peak and fall back to zero:
- The area under the Force vs. Time curve is equal to the delivered Impulse (J = ∫ F dt).
- A triangular profile is often used to approximate simple collisions, where: J = 1/2 * t_contact * F_peak.
- Therefore, the peak force reached is exactly double the average force: F_peak = 2 * J / t_contact.
Solved Examples
A tennis player strikes a stationary 0.06 kg tennis ball with an average force of 240 N. The contact duration between the racket and the ball is 0.005 seconds (5 ms). Calculate the impulse applied to the ball and its final velocity.
- First, calculate the applied impulse: J = F * t = 240 N * 0.005 s = 1.2 N·s (or kg·m/s).
- By the Impulse-Momentum Theorem, Impulse is equal to the change in momentum: J = Δp = m * v_f - m * v_i.
- Since the ball starts from rest, initial velocity v_i = 0, so: J = m * v_f.
- Solve for final velocity: v_f = J / m = 1.2 kg·m/s / 0.06 kg = 20 m/s.
- The impulse is 1.2 N·s and the ball's final velocity is 20 m/s.
Answer: Impulse: 1.2 N·s | Final Velocity: 20 m/s
A 1,200 kg car travelling at 15 m/s crashes into a wall. If it hits a concrete barrier, it stops in 0.1 seconds. If it hits a water-filled cushion barrier, it stops in 0.5 seconds. Compare the average impact force experienced by the car in both cases.
- The initial momentum of the car: p_i = m * v_i = 1,200 kg * 15 m/s = 18,000 kg·m/s.
- The final momentum of the car: p_f = 0 (since it stops).
- The change in momentum for both crashes is identical: Δp = p_f - p_i = -18,000 kg·m/s. Thus, the required stopping impulse is J = -18,000 N·s.
- For the concrete barrier (t = 0.1 s): F_avg = J / t = -18,000 N·s / 0.1 s = -180,000 N (or 180 kN).
- For the water cushion barrier (t = 0.5 s): F_avg = J / t = -18,000 N·s / 0.5 s = -36,000 N (or 36 kN).
- By extending the crash time by 5 times, the safety cushions reduce the average impact force to 1/5th of its original severity, saving lives.
Answer: Concrete: 180,000 N | Safety Cushions: 36,000 N
A constant thrust force of 60 N is applied to a 15 kg space probe initially at rest. If the thruster fires for exactly 4.0 seconds, find the impulse applied and the final speed of the probe.
- The impulse applied by the thruster: J = F * t = 60 N * 4.0 s = 240 N·s.
- According to the Impulse-Momentum Theorem: J = Δp = m * v_f - m * v_i.
- Since the probe starts from rest (v_i = 0), the change in velocity is: v_f = J / m.
- Substitute the values: v_f = 240 N·s / 15 kg = 16 m/s.
- The delivered impulse is 240 N·s and the probe reaches a speed of 16 m/s.
Answer: Impulse: 240 N·s | Speed: 16 m/s
Common Misconceptions
- "Impulse and force are the same thing": Force is an instantaneous push/pull. Impulse is force accumulated over a period of time. An enormous force applied for zero time delivers zero impulse.
- "Elastic bounces experience less force": Bouncing back (rebound) involves a larger change in velocity, meaning the impulse is up to double that of a sticky collision. Thus, bouncing off a wall hurts more than sticking to it.
- "SI units are different": 1 N·s = 1 kg·m/s. They are completely interchangeable and represent the same physical dimensions.
Quick Summary
- Impulse is force multiplied by contact duration: J = F * t.
- Impulse is equal to change in momentum: J = Δp.
- SI Unit is N·s or kg·m/s.
- Extending contact time t reduces the peak force F_peak.
- The area under a Force-Time curve represents the delivered Impulse.
Practice Questions
1. What are the SI units of impulse, and how do they relate to momentum units?
The standard SI unit for impulse is the Newton-second (N·s). Since a Newton is 1 kg·m/s², multiplying by seconds gives kg·m/s, which is exactly the unit of momentum. Therefore, 1 N·s is dimensionally identical to 1 kg·m/s.
2. A soccer player kicks a 0.45 kg ball with a force of 150 N. If the boot is in contact with the ball for 0.02 seconds, what is the change in the ball's momentum?
The change in momentum is equal to the impulse: Δp = J = F * t. Substituting the values: Δp = 150 N * 0.02 s = 3.0 kg·m/s (or N·s).
3. Why do golfers and batters try to 'follow through' with their swing?
Following through increases the contact time (t) of the club or bat with the ball. Since Impulse = Force * time (J = F * t), increasing contact time for a given force increases the total impulse delivered, resulting in a larger change in momentum and a faster, longer hit.
4. If a rubber ball bounces elastically off a wall, is the impulse greater than if a clay ball of the same mass sticks to the wall?
Yes. For a clay ball that sticks, the change in velocity is Δv = 0 - v_i = -v_i. For an elastically bouncing ball, it rebounds, so Δv = -v_i - v_i = -2v_i. The momentum change and therefore the impulse for the bouncing ball is twice as large as the sticking ball.
FAQ
Frequently Asked Questions
What is impulse?
Impulse is a vector quantity defined as the product of the net force acting on an object and the time duration of that force: J = F * t. It describes how force changes an object's momentum.
What is the Impulse-Momentum Theorem?
It states that the impulse applied to an object is exactly equal to its change in momentum: J = Δp = m * Δv. It is derived directly from Newton's Second Law (F = Δp/Δt).
How does padding in helmets reduce injuries?
When a player experiences an impact, their change in momentum is fixed. Soft padding increases the duration of contact (t). Since J = F * t, a larger contact duration (t) reduces the peak impact force (F) acting on the head.
Can impulse be negative?
Yes. Since impulse is a vector quantity, its direction matches the force. If the force acts in the negative direction (e.g. opposite to initial motion to slow an object down), the impulse is negative.
What does the area under a Force-Time graph represent?
The area under a Force vs. Time graph represents the total impulse delivered. For variable forces, this area equals the integration of force over time: J = ∫ F dt.
What is the SI unit of impulse?
The SI unit of impulse is newton-second, written as N·s. It is equivalent to kg·m/s, the unit of momentum.
How can impulse increase the speed of a ball?
A larger impulse produces a larger change in momentum. You can increase impulse by applying a bigger force, increasing contact time, or both.
Why does follow-through matter in sports?
Follow-through keeps the force acting for a slightly longer time, increasing impulse and causing a greater change in momentum.
Is impulse scalar or vector?
Impulse is a vector because force is a vector. The direction of impulse is the direction of the net force.
How is impulse different from force?
Force is a push or pull at an instant, while impulse measures the effect of force acting over a time interval.
Can a small force create a large impulse?
Yes. A small force can create a large impulse if it acts for a long enough time.