The velocity versus time graph of a body moving along a straight path;
(i) Calculate the distance covered by the body from
t = 0 to t= 8 s.
(ii) Find its net displacement from t = 0 to 8 s.
Answers
Answer:
Step-by-step explanation:
Motion problems are very common throughout calculus. In differential calculus, we reasoned about a moving object's velocity given its position function. In integral calculus we go in the opposite direction: given the velocity function of a moving object, we reason about its position or about the change in its position.
Thinking about velocity, speed, and definite integrals
Say a particle moves in a straight line with velocity v(t)=5-tv(t)=5−tv, left parenthesis, t, right parenthesis, equals, 5, minus, t meters per second, where ttt is time in seconds.
When the velocity is positive it means the particle is moving forward along the line, and when the velocity is negative it means the particle is moving backwards.
Say we are asked for the particle's displacement (i.e. the change in its position) between t=0t=0t, equals, 0 seconds and t=10t=10t, equals, 10 seconds. Since the velocity is the rate of change of the particle's position, any change in the position of the particle is given by a definite integral.
Specifically, we are looking for \displaystyle\int_0^{10} v(t)\,dt∫
0
10
v(t)dtintegral, start subscript, 0, end subscript, start superscript, 10, end superscript, v, left parenthesis, t, right parenthesis, d, t.
Interestingly, the displacement is \displaystyle\int_0^{10} v(t)\,dt=0∫
0
10
v(t)dt=0integral, start subscript, 0, end subscript, start superscript, 10, end superscript, v, left parenthesis, t, right parenthesis, d, t, equals, 0 meters. (You can see how the two areas in the graph are equal in size and opposite in sign).
The displacement being 000 means that the particle was at the same position at times t=0t=0t, equals, 0 and t=10t=10t, equals, 10 seconds. This makes sense when you see how the particle first moves forwards and then backwards, so it gets back to where it started.