Question: A proton moves along the axis according to the equation x = 50t + 10t2 , where x is in meters and t is in seconds. Calculate (a) the average velocity of the proton during the first 3.0 s of its motion, (b) the instantaneous velocity of the proton at t = 3.0 s, and (c) the instantaneous acceleration of the proton at t = 3.0 s . (d) Graph x versus t and indicate how the answer to (a) can be obtained from the plot. (e) Indicate the answer to (b) on the graph. (f) Plot v versus t and indicate on it the answer to (c).
The equation for the motion of the proton is,
Average velocity is the ratio of total displacement to the total time interval. Instantaneous velocity is the velocity of a moving object at a specific moment.
The expression for the average velocity is given as follows:
Here, is the displacement and is the time duration.
The expression for the instantaneous velocity is given as follows:
The expression for the instantaneous acceleration is given as follows:
Position of the proton at is,
Position of the proton at is,
Using equation (i), the average velocity is calculated as follows:
Thus, the average velocity of the proton is .
Using equation (ii), the instantaneous velocity is,
The instantaneous velocity at is,
Thus, the instantaneous velocity at time is .
Using equation (iii), the instantaneous acceleration is,
Thus, the instantaneous acceleration at is .
The graph x vs t is plotted below.
From the graph,
The positions at 3.0 s and 0 s are,
So, the average velocity is,
The instantaneous velocity at is the slope of the triangle drawn at that point in the above graph and it is,
Thus, the instantaneous velocity at 3.0 s is 110 m/s.
The graph of vs is as below,
From the graph, the instantaneous acceleration is the slope of the graph
Thus, the instantaneous acceleration from the graph is .
A ball is thrown down vertically with an initial speed of from a height of . (a) What is its speed just before it strikes the ground? (b) How long does the ball take to reach the ground? What would be the answers to (c) part a and (d) part b if the ball were thrown upward from the same height and with the same initial speed? Before solving any equations, decide whether the answers to (c) and (d) should be greater than, less than, or the same as in (a) and (b).
A train started from rest and moved with constant acceleration. At one time, it was travelling30 m/s , and 160 m farther on it was travelling 50 m/s. Calculate (a) the acceleration (b) the time required to travel 160 m mentioned (c) the time required to attain speed of 30 m/s (d) the distance moved from rest to the time the train had a speed of 30 m/s (e) Graph x vs t and v vs t for the train, from rest.
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