REPORT AIM The aim of this experiment is to: ? Explore the equations of uniform accelerated motion and investigate the relationship between displacement and time ? Determine the magnitude of deceleration due to friction. ? Assess the effect of mass on the car’s accelerated motion. DESIGN Hypothesis – A car moving in a straight line with a non-zero initial velocity will finally come to a rest as a result of friction, given that the car has no engine or external tractions. This motion can be considered as a uniform accelerated motion because: 1.
The car is moving in a horizontal straight line so weight is cancelled by the normal reaction force from the ground. The only other force existed is the friction between the car and the surface therefore it will be equal to the net force 2. According to the formula Fr = ? N, the amount of the friction depends on two factors: the friction coefficient and the normal reaction force, both of which are fixed. Therefore the amount of friction is constant throughout the motion 3. According to Newton’s Second Law F = ma, a constant net force will result in a uniform acceleration (deceleration).
The acceleration is negative in this case as cars are slowing down to a rest. For convenience, this decelerated motion can be inverted into an equivalent motion in which the car is acceleration from rest. It should follow the equation of x = ut + 1/2 at2, where x = distance traveled, u = 0 (seen as the initial velocity but actually is the final velocity which is zero at rest), a = acceleration (actually deceleration) and t = time taken during that motion. This formula can be simplified as x = 1/2 at2.
We will measure the variables of x and t to verify this relationship and determine the magnitude of this deceleration, which can be derived from the gradient of the regression line of x against t2. Theoretically the mass of the car should not influence its deceleration because: Friction is the net force, Fr =? N = ? mg (as N=Weight) = Fnet = ma => ? mg = ma => a = ? g, which implies deceleration depends only on the friction coefficient. The car’s initial velocity at the beginning of the horizontal tract is provided by releasing the car from a ramp connected to the tract.
This design will reduce human interventions to the car’s horizontal part of motion. Three cars of different masses will be tested to see if mass have any effect on the magnitude of deceleration. METHODS A plastic track is connected to a ramp which has a height of 9cm and contact surface of 10cm. A car is hold still on the ramp surface at a position of 3cm from the top (i. e. it will travel 7 cm to reach the bottom of the ramp). Three different cars are used in this experiment and have masses of 0. 231kg, 0. 358kg and 0. 535kg respectively. All of the three cars are released from the same position.
Due to gravity, the car will gain a velocity when entering the horizontal tract. By using a stopwatch, we take measurement of the time from the moment when the car’s rear wheels first reach the horizontal tract and stop the measurement as soon as it comes to rest. The distance traveled by the car along the plastic tract is measured by a hard ruler and distance is taken from the back of the car’s rear wheels (once stopped) to the junction between the ramp and the plastic tract. [pic] This experiment is repeated three times and all data are entered into excel.
A graph of distance against time squared is plotted and a regression analysis is performed to determine the relationship between the two variables and the gradient of the line. RESULTS The results of the experiment are summarized in the following table: | |Distance x (m) |Time (s) |t^2 | |Exp1 Car1 |0. 235 |0. 78 |0. 6084 | |Exp1 Car2 |0. 285 |1. 73 |2. 9929 | |Exp1 Car3 |0. 403 |2. |5. 29 | |Exp2 Car1 |0. 287 |1. 3 |1. 69 | |Exp2 Car2 |0. 308 |1. 82 |3. 3124 | |Exp2 Car3 |0. 335 |2. 03 |4. 1209 | |Exp3 Car1 |0. 104 |0. 63 |0. 3969 | |Exp3 Car2 |0. 255 |1. 54 |2. 3716 | |Exp3 Car3 |0. 374 |1. 98 |3. 9204 |
Car 1 – 0. 231kg Car 2 – 0. 358kg Car 3 – 0. 535kg The Graph of distance against time squared: [pic] The gradient of the regression line is equal to 1/2 a, so a = 2 x 0. 0479 = 0. 0958 ms-2. The uncertainty for time measurement is estimated to be 10% and 1% for distance measurement (±2mm). As a = 2x/(t2), the uncertainty percentage for a is approximately = 1 % + 10% + 10% = 21%. Therefore a = 0. 0958±0. 0201 ms-2. DISCUSSION From the above graph, it is evident that there is a strong linear positive relationship between distance and time squared (R2 = 0. 8067).
The distance traveled by the car during deceleration is proportional to the square of time. This is consistent with our hypothesis that x = 1/2 at2. The gradient of the regression line is equal to 1/2 a. Thus, deceleration = 2 x gradient = 2 x 0. 0479 = 0. 0958 ms-2. A R2 value of 0. 8067 means nearly 81% of the variation in distance can be explained by variation in time, according to the linear model of x against t2. This suggests that mass of the car is not a factor determining the magnitude of acceleration and does not influence the pattern of this decelerated motion.
There are a few factors in this experiment which may cause variations to the result. For example, we assume the friction coefficient of the plastic tract is the same for the three cars thus they will have the same deceleration, as discussed in Experimental Design. However, the tyres of the three cars might be made from different materials so the friction coefficients might be slightly different. As a result, the deceleration is not actually constant and this creates error for our linear regression. Another compounding a factor is the joint between the ramp and the horizontal tract.
As the car passes through this point, the direction of the movement may be slightly diverted due to this sharp turn. Other issues affecting experimental results include not-leveled tract, air resistance, and human errors in measurements such as delayed timing. This experiment can be improved by eliminating these compounding factors. For example, we could use a spirit level to check the plastic tract and ensure that cars are moving in a horizontal plane. The cars’ tyres should be made of the same material. The bottom of the ramp should be connected to the plastic track smoothly without any sharp corners.
More measurements are to be taken to increase the accuracy of data. CONCLUSION From this experiment, we successfully verified the equation for a uniform decelerated motion x = 1/2 at2. Our results showed that distance was proportional to the square of time. The deceleration was determined from the gradient of the regression line of x against t2, which was found to be 0. 0958±0. 0201 ms-2. The mass of the car has no effect on the value of deceleration and the pattern of motion. The accuracy of the results can be improved by eliminating compounding factors such as different tyre material and tract surface layout.