EQUATIONS OF CONSTANT ACCELERATION LEARNING GOALS Students will Derive the five key equations of accelerated motion apply to motion with uniform (constant) acceleration. Select which equation(s) to use based on the given and required variables of the problem. PREPARATION AT HOME Reading Nelson Physics 11 Section 1.5 Pg.36-39 Complete the following derivations on page 1-2 of this note and check your answers. Reading Quiz RECALL: DEFINITIONS The average velocity,, of a moving object is defined as its displacement over a specific interval of time. = Δ Δ = Δ The average acceleration,, of a moving object is defined as the change in velocity of a specific interval of time. = Δ Δ = Δ As long as the motion produces a graph that is straight, the midpoint of the velocities gives the average velocity. = + 2 Using these definitions (also called first principles), we can derive several equations that will help us determine aspects of an object s motion when undergoing constant acceleration. UNIFORM (CONSTANT) ACCELERATION Consider the following velocity vs. Time graph: What type of motion is shown? (1) (2) (3) This is critical for all the equations you are about to derive 1
Equation 1: Finding displacement when you know the acceleration, initial velocity and time, but don t know the final velocity Strategy: Calculate the area under the graph as a rectangle plus a triangle = lw + ½ bh Now sub in the length, width, base and height from the graph (i.e. v1, v2, etc.) But we don t have v 2 so use the equation for acceleration = and rearrange it to make a convenient substitution. Now sub it into equation (1) and simplify it. Check your final equation with the teacher. Equation 2: Finding displacement when you know the acceleration, final velocity and time, but don t know the initial velocity Strategy: Calculate the area under the graph as a big rectangle subract a triangle = lw - ½ bh Now sub in the length, width, base and height from the graph (i.e. v1, v2, etc.) Follow the same procedure as for equation 1 and then simplify. Check your final equation with the teacher. 2
Equation 3: Finding displacement when you know the initial and final velocity and the time but not the acceleration. Strategy: Calculate the area under the graph as a rectangle. = lw Now sub in the length and width, from the graph (i.e. v1, v2, etc.) Simplify your equation if necessary and check it with the teacher. Equation 4: Finding final velocity when you know initial velocity, displacement and acceleration but not the time You want to derive an equation = +2. Start with one of the others and sub in. Good luck! (Use your own paper) To check your answers, look at pg 37. You should find all 5 equations there. Check each one of yours carefully and if you find a mistake, figure out where you went wrong and fix it. Each of the following examples is setup for you to use the GUESS method to solve. The power of this method is apparent when you need to choose the needed equation. Look at your givens and select the equation that has those givens AND the unknown you are trying to find. EXAMPLE 1: DISPLACEMENT OVER AN ACCELERATED PERIOD OF TIME A Ferrari moving at 20 km/h accelerates to 230 km/h in 7.50 s. What distance does it cover in doing so? What is its acceleration? Diagram Given = = Δ= Equation & Solution Statement Unknown Δ=? =? 3
EXAMPLE 2: SLOWING DOWN TO A STOP If the same Ferrari runs out of gas while traveling at 230 km/h, the driver can put it into neutral and coast 710 m before coming to a stop. Find the acceleration of the Ferrari as it slows to a stop and find the time it takes. Diagram Given = = Δ= Equation & Solution Statement Unknown =? Δ=? PRACTICE PROBLEMS 1. A hybrid car with an initial velocity of 10.0 m/s [E] accelerates at 3.0 m/s 2 [E]. How long will it take the car to acquire a final velocity of 25.0 m/s [E]? 2. A coal train travelling at 16.0 m/s is brought to rest in 8.0 s. Find the distance travelled by the coal train while it is coming to a stop. Assume uniform acceleration. 3. A golf ball that is initially travelling at 25 m/s hits a sand trap and slows down with an acceleration of -20 m/s 2. Find its displacement after 1.0 s. 4. A speedboat slows down at a rate of 5.0 m/s 2 and comes to a stop. If the process took 15 s, find the displacement of the boat. 5. A bullet accelerates the length of the barrel of a rifle (0.750 m) with a magnitude of 5.35 x 10 5 m/s 2. With what speed does the bullet exit the barrel? 6. How far will a humanoid robot travel in 3.0 s, accelerating at 1.0 cm/s 2 [forward], if its initial velocity is 5.0 cm/s [forward]? 7. What is the displacement of a logging truck accelerating from 10 m/s [right] to 20 m/s [right] in 5.0 s? 8. Leeran is driving at a constant speed of 55 km/h. He sees that the traffic light 60 metres ahead of him turns yellow, so he floors it in order to beat the light. If his acceleration was constant at 4 m/s 2, will he be able to beat the yellow light which lasts for 3.0 seconds? Answers: 1. 5.0 s 2. 64 m 3. 15 m [fwd] 4. 563 m [fwd] 5. 896 m/s 6. 19.5 cm 7. 75 m [right] 8. 63.8 m (yes) 4
SPH3U1 Lesson 09 Kinematics KINEMATICS PROBLEM SOLVING 1. How far will a car travel if it starts from rest and experiences an acceleration of magnitude 3.75 m/s 2 [forward] for 5.65 s? [60. m] 2. Determine the acceleration of a bullet starting from rest and leaving the muzzle 2.75 x 10-3 s later with a velocity of 460 m/s [forward]. [1.7 x 10 5 m/s 2 ] 3. A car is travelling on the highway at 105 km/h. The driver wants to pass someone so he accelerates at a rate of 3.0 (km/h)/s for a time of 5.0 seconds. What is the car's final speed? [120 km/h] 4. An elk moving at a velocity of 20 km/h [N] accelerates at 1.5 m/s 2 [N] for 9.3 s until it reaches its maximum velocity. Calculate its maximum velocity in km/h. [70. km/h] 5. Determine the magnitude of a car's acceleration if its stopping distance is 39.0 m for an initial speed of 97.0 km/h. [9.3 m/s 2 ] 6. A ball moves up a hill with an initial velocity of 3.0 m/s. Four seconds later, it is moving down the hill at 9.0 m/s. Find the displacement of the ball from its initial point of release. [12 m down the hill] 7. A subway train starts from rest at a station and accelerates at the rate of 2.0 m/s 2 [W] for 10. s. It runs at a constant speed for the next 30.0 s and then decelerates at 2.4 m/s 2 until it stops at the next station. Find the total distance between the stations and the average speed of the train. 8. On-ramps are designed so that motorists can move seamlessly into highway traffic. If a car needs to increase its speed from 50 km/h to 100 km/h and the engine can provide a maximum acceleration of magnitude 3.8 m/s 2, find the minimum length of the on-ramp. [76 m] 9. A motorcycle ride consists of two segments. During the first segment, the motorcycle starts from rest, has an acceleration of 2.6 m/s 2 [E] and a displacement of 120 m [E]. Immediately after the first segment the motorcycle enters the second segment and begins slowing down with an acceleration of 1.5 m/s 2 [W] until its velocity is 12 m/s [E]. What is the displacement of the motorcycle during the second segment. 10. A skier starting from rest accelerates uniformly downhill at 1.8 m/s 2 [forward]. How long will it take the skier to reach a point 95 m [forward] from the stating position? [10. s] 1